💡Okay, so I need to find the maximum y-coordinate of a point on the graph of r = sin(3θ). Hmm, I remember that in polar coordinates, the y-coordinate can be expressed in terms of r and θ. Let me recall the conversion formulas from polar to Cartesian coordinates. I think y is equal to r times sinθ, right? So, y = r sinθ.Given that r = sin(3θ), I can substitute that into the equation for y. So, y = sin(3θ) * sinθ. Now, I need to simplify this expression to make it easier to find the maximum value. Maybe I can use some trigonometric identities here. I remember that sin(3θ) can be expanded using the triple angle formula. Let me write that down: sin(3θ) = 3sinθ - 4sin³θ. So, substituting that into y, I get y = (3sinθ - 4sin³θ) * sinθ. Multiplying through, that becomes y = 3sin²θ - 4sin⁴θ. Hmm, okay, so y is a function of sinθ. Maybe I can let u = sinθ to make it simpler. Then, y = 3u² - 4u⁴. Now, to find the maximum value of y, I need to find the critical points of this function. That means taking the derivative of y with respect to u and setting it equal to zero. Let's compute dy/du. The derivative of 3u² is 6u, and the derivative of -4u⁴ is -16u³. So, dy/du = 6u - 16u³. Setting this equal to zero gives 6u - 16u³ = 0. I can factor out a 2u from this equation: 2u(3 - 8u²) = 0. So, the critical points occur when 2u = 0 or 3 - 8u² = 0. Solving 2u = 0 gives u = 0. Solving 3 - 8u² = 0 gives 8u² = 3, so u² = 3/8, which means u = ±√(3/8). Simplifying √(3/8), that's equal to √6 / 4, right? Because √(3/8) = √6 / (2*2) = √6 / 4. So, the critical points are u = 0 and u = ±√6 / 4. Now, I need to determine which of these gives the maximum y-coordinate. Let's evaluate y at each critical point.First, when u = 0: y = 3(0)² - 4(0)⁴ = 0. So, y = 0 here.Next, when u = √6 / 4: Let's compute y. y = 3u² - 4u⁴. First, compute u²: (√6 / 4)² = 6 / 16 = 3/8.Then, u⁴ = (u²)² = (3/8)² = 9/64.So, plugging back into y: y = 3*(3/8) - 4*(9/64) = 9/8 - 36/64.Simplify 36/64: that's 9/16.So, y = 9/8 - 9/16. To subtract these, convert 9/8 to 18/16. So, 18/16 - 9/16 = 9/16.Wait, that can't be right. Let me double-check my calculations. Wait, hold on. I think I made a mistake in the substitution. Let me go back.Wait, y = 3u² - 4u⁴, and u = √6 / 4.So, u² = 6/16 = 3/8.u⁴ = (3/8)² = 9/64.So, y = 3*(3/8) - 4*(9/64) = 9/8 - 36/64.Convert 9/8 to 72/64, so 72/64 - 36/64 = 36/64 = 9/16.Hmm, so y = 9/16 when u = √6 / 4.Wait, but earlier when I used a different method, I got 4√3 / 9, which is approximately 0.7698, while 9/16 is 0.5625. These are different. Did I do something wrong here?Wait, maybe I messed up the substitution. Let me check my steps again.Wait, the original expression for y was y = sin(3θ) sinθ. Then I used the triple angle formula: sin(3θ) = 3sinθ - 4sin³θ. So, y = (3sinθ - 4sin³θ) sinθ = 3sin²θ - 4sin⁴θ. That seems correct.Then, I set u = sinθ, so y = 3u² - 4u⁴. Taking derivative: dy/du = 6u - 16u³. Setting to zero: 6u - 16u³ = 0, which factors to 2u(3 - 8u²) = 0. So, u = 0 or u² = 3/8, so u = ±√(3/8) = ±√6 / 4. That seems correct.Then, plugging back into y: y = 3u² - 4u⁴.So, u² = 3/8, u⁴ = 9/64.So, y = 3*(3/8) - 4*(9/64) = 9/8 - 36/64.Convert 9/8 to 72/64, so 72/64 - 36/64 = 36/64 = 9/16. So, y = 9/16.But wait, earlier I thought the maximum y-coordinate was 4√3 / 9, which is approximately 0.7698, but 9/16 is 0.5625. These are different. So, which one is correct?Wait, maybe I made a mistake in the initial substitution. Let me try a different approach.Alternatively, I can express y = sin(3θ) sinθ and use product-to-sum formulas.Recall that sinA sinB = [cos(A - B) - cos(A + B)] / 2.So, y = sin(3θ) sinθ = [cos(2θ) - cos(4θ)] / 2.So, y = (cos2θ - cos4θ)/2.Now, to find the maximum value of y, we can analyze this expression.Let me denote f(θ) = (cos2θ - cos4θ)/2.To find the maximum of f(θ), we can take the derivative with respect to θ and set it to zero.Compute f'(θ):f'(θ) = [ -2sin2θ + 4sin4θ ] / 2 = (-2sin2θ + 4sin4θ)/2 = -sin2θ + 2sin4θ.Set f'(θ) = 0:-sin2θ + 2sin4θ = 0.Let me recall that sin4θ = 2sin2θ cos2θ. So, substitute that in:-sin2θ + 2*(2sin2θ cos2θ) = 0.Simplify:-sin2θ + 4sin2θ cos2θ = 0.Factor out sin2θ:sin2θ(-1 + 4cos2θ) = 0.So, either sin2θ = 0 or -1 + 4cos2θ = 0.Case 1: sin2θ = 0.This implies 2θ = nπ, so θ = nπ/2, where n is an integer.Case 2: -1 + 4cos2θ = 0 => 4cos2θ = 1 => cos2θ = 1/4.So, 2θ = ± arccos(1/4) + 2πn, so θ = ± (1/2) arccos(1/4) + πn.Now, let's evaluate f(θ) at these critical points.First, for Case 1: θ = nπ/2.Compute f(θ) = (cos2θ - cos4θ)/2.If θ = 0: f(0) = (1 - 1)/2 = 0.If θ = π/2: f(π/2) = (cosπ - cos2π)/2 = (-1 - 1)/2 = -1.Similarly, other multiples will give either 0 or -1, so the maximum here is 0.Case 2: θ = ± (1/2) arccos(1/4) + πn.Let me compute f(θ) at θ = (1/2) arccos(1/4).Let me denote φ = arccos(1/4). So, cosφ = 1/4, and sinφ = √(1 - (1/4)²) = √(15/16) = √15 / 4.Then, θ = φ/2.Compute f(θ) = (cos2θ - cos4θ)/2.First, compute cos2θ and cos4θ.cos2θ = cosφ = 1/4.cos4θ = cos(2φ) = 2cos²φ - 1 = 2*(1/4)² - 1 = 2*(1/16) - 1 = 1/8 - 1 = -7/8.So, f(θ) = (1/4 - (-7/8))/2 = (1/4 + 7/8)/2.Convert 1/4 to 2/8: (2/8 + 7/8)/2 = (9/8)/2 = 9/16.So, f(θ) = 9/16 at θ = (1/2) arccos(1/4). Similarly, at θ = -(1/2) arccos(1/4), the value will be the same because cosine is even.So, the maximum value of y is 9/16.Wait, but earlier I thought it was 4√3 / 9. Which one is correct?Wait, let me compute 4√3 / 9 numerically: √3 ≈ 1.732, so 4*1.732 ≈ 6.928, divided by 9 is approximately 0.7698.9/16 is 0.5625, which is less than 0.7698. So, which one is correct?Wait, maybe I made a mistake in the first method. Let me check.In the first method, I expressed y = sin(3θ) sinθ, then used the triple angle formula to get y = 3sin²θ - 4sin⁴θ, then set u = sinθ, so y = 3u² - 4u⁴. Then, took derivative dy/du = 6u - 16u³, set to zero, got u = 0 or u² = 3/8, so u = ±√6 / 4.Then, plugging back, y = 3*(3/8) - 4*(9/64) = 9/8 - 36/64 = 9/8 - 9/16 = (18/16 - 9/16) = 9/16.So, that seems correct.But in the second method, using product-to-sum, I also got y = 9/16.Wait, but earlier I thought the maximum was 4√3 / 9. Maybe I confused it with another problem.Wait, let me check another approach. Maybe using calculus in terms of θ.Express y = sin(3θ) sinθ.Let me write y = sin(3θ) sinθ.To find the maximum, take derivative dy/dθ and set to zero.Compute dy/dθ:dy/dθ = d/dθ [sin3θ sinθ] = 3cos3θ sinθ + sin3θ cosθ.Set this equal to zero:3cos3θ sinθ + sin3θ cosθ = 0.Let me factor this expression.Hmm, maybe use trigonometric identities.Note that sin3θ = 3sinθ - 4sin³θ, and cos3θ = 4cos³θ - 3cosθ.But that might complicate things. Alternatively, perhaps factor terms.Let me factor sinθ:sinθ (3cos3θ + (sin3θ / sinθ) cosθ) = 0.Wait, that might not be helpful.Alternatively, let me write the equation as:3cos3θ sinθ + sin3θ cosθ = 0.Notice that this resembles the sine addition formula: sin(A + B) = sinA cosB + cosA sinB.But here, we have 3cos3θ sinθ + sin3θ cosθ. Hmm, not exactly the same.Wait, let me see: 3cos3θ sinθ + sin3θ cosθ = 0.Let me factor out sinθ:sinθ (3cos3θ + (sin3θ / sinθ) cosθ) = 0.But sin3θ / sinθ is 3 - 4sin²θ, from the identity sin3θ = 3sinθ - 4sin³θ.So, sin3θ / sinθ = 3 - 4sin²θ.Thus, the equation becomes:sinθ [3cos3θ + (3 - 4sin²θ) cosθ] = 0.So, either sinθ = 0, which gives y = 0, or the other factor is zero:3cos3θ + (3 - 4sin²θ) cosθ = 0.Let me simplify this expression.First, expand the terms:3cos3θ + 3cosθ - 4sin²θ cosθ = 0.Now, let me express cos3θ in terms of cosθ. Recall that cos3θ = 4cos³θ - 3cosθ.So, substitute that in:3(4cos³θ - 3cosθ) + 3cosθ - 4sin²θ cosθ = 0.Expand:12cos³θ - 9cosθ + 3cosθ - 4sin²θ cosθ = 0.Combine like terms:12cos³θ - 6cosθ - 4sin²θ cosθ = 0.Factor out cosθ:cosθ (12cos²θ - 6 - 4sin²θ) = 0.So, either cosθ = 0, which would give θ = π/2 + nπ, or the other factor is zero:12cos²θ - 6 - 4sin²θ = 0.Let me express everything in terms of cosθ. Recall that sin²θ = 1 - cos²θ.So, substitute:12cos²θ - 6 - 4(1 - cos²θ) = 0.Simplify:12cos²θ - 6 - 4 + 4cos²θ = 0.Combine like terms:(12cos²θ + 4cos²θ) - 10 = 0 => 16cos²θ - 10 = 0.So, 16cos²θ = 10 => cos²θ = 10/16 = 5/8 => cosθ = ±√(5/8) = ±√10 / 4.So, θ = arccos(√10 / 4) or θ = arccos(-√10 / 4), etc.Now, let's compute y at these θ values.First, let's take θ = arccos(√10 / 4). Let's denote φ = arccos(√10 / 4). So, cosφ = √10 / 4, and sinφ = √(1 - (10/16)) = √(6/16) = √6 / 4.Compute y = sin(3φ) sinφ.First, compute sin3φ. Using the identity sin3φ = 3sinφ - 4sin³φ.So, sin3φ = 3*(√6 / 4) - 4*(√6 / 4)^3.Compute each term:3*(√6 / 4) = 3√6 / 4.(√6 / 4)^3 = (6√6) / 64 = (3√6) / 32.So, 4*(√6 / 4)^3 = 4*(3√6 / 32) = 12√6 / 32 = 3√6 / 8.Thus, sin3φ = 3√6 / 4 - 3√6 / 8 = (6√6 / 8 - 3√6 / 8) = 3√6 / 8.Now, sinφ = √6 / 4.So, y = sin3φ sinφ = (3√6 / 8) * (√6 / 4) = (3*6) / (32) = 18 / 32 = 9 / 16.Similarly, for θ = arccos(-√10 / 4), we would get the same value because sin(3θ) would be negative, but sinθ would also be negative, so their product would still be positive 9/16.So, the maximum y-coordinate is 9/16.Wait, but earlier I thought it was 4√3 / 9. Maybe I confused it with another problem. Let me check numerically.Compute 9/16: 9 divided by 16 is 0.5625.Compute 4√3 / 9: √3 ≈ 1.732, so 4*1.732 ≈ 6.928, divided by 9 is approximately 0.7698.So, 9/16 is less than 4√3 / 9. Which one is correct?Wait, maybe I made a mistake in the first method. Let me check.In the first method, I expressed y = sin(3θ) sinθ, then used the triple angle formula to get y = 3sin²θ - 4sin⁴θ, then set u = sinθ, so y = 3u² - 4u⁴. Then, took derivative dy/du = 6u - 16u³, set to zero, got u = 0 or u² = 3/8, so u = ±√6 / 4.Then, plugging back, y = 3*(3/8) - 4*(9/64) = 9/8 - 36/64 = 9/8 - 9/16 = (18/16 - 9/16) = 9/16.So, that seems correct.But in the second method, using product-to-sum, I also got y = 9/16.Wait, but in the initial problem, the user had a different approach, leading to 4√3 / 9. Maybe that was incorrect.Wait, let me check the initial problem again.Wait, the user wrote:"For r = sin3θ,y = r sinθ = sin3θ sinθ = (sin2θ cosθ + cos2θ sinθ) sinθ = ... = 2sinθ - 2sin³θ.Let u = sinθ. Then y = 2u - 2u³. Taking derivative, y' = 2 - 6u² = 0 => u² = 1/3 => u = ±√3 / 3.Thus, y = 2*(√3 / 3) - 2*(√3 / 3)^3 = 2√3 / 3 - 2*(3√3)/27 = 2√3 / 3 - 2√3 / 9 = (6√3 - 2√3)/9 = 4√3 / 9."Wait, so in this approach, they expanded sin3θ as sin(2θ + θ) and used the sine addition formula, leading to y = 2sinθ - 2sin³θ.Then, setting u = sinθ, y = 2u - 2u³, derivative y' = 2 - 6u², set to zero, u² = 1/3, u = ±√3 / 3.Then, y = 2*(√3 / 3) - 2*(√3 / 3)^3 = 2√3 / 3 - 2*(3√3)/27 = 2√3 / 3 - 2√3 / 9 = (6√3 - 2√3)/9 = 4√3 / 9.So, according to this method, the maximum y is 4√3 / 9.But in my methods, I got 9/16. So, which one is correct?Wait, let me compute both values numerically:4√3 / 9 ≈ 4*1.732 / 9 ≈ 6.928 / 9 ≈ 0.7698.9/16 = 0.5625.So, 4√3 / 9 is larger than 9/16. So, which one is the actual maximum?Wait, maybe I made a mistake in my second method. Let me check.In the second method, I used product-to-sum and got y = (cos2θ - cos4θ)/2. Then, I found critical points by taking derivative and got y = 9/16.But in the first method, the user got 4√3 / 9. So, which one is correct?Wait, perhaps I made a mistake in the substitution when using the product-to-sum formula.Wait, let me re-express y = sin3θ sinθ.Using product-to-sum: sinA sinB = [cos(A - B) - cos(A + B)] / 2.So, y = [cos(2θ) - cos(4θ)] / 2.Yes, that's correct.Then, f(θ) = (cos2θ - cos4θ)/2.Taking derivative: f’(θ) = [ -2sin2θ + 4sin4θ ] / 2 = -sin2θ + 2sin4θ.Set to zero: -sin2θ + 2sin4θ = 0.Express sin4θ as 2sin2θ cos2θ: -sin2θ + 2*(2sin2θ cos2θ) = -sin2θ + 4sin2θ cos2θ = 0.Factor: sin2θ(-1 + 4cos2θ) = 0.So, sin2θ = 0 or cos2θ = 1/4.When sin2θ = 0, θ = 0, π/2, etc., giving y = 0.When cos2θ = 1/4, then 2θ = ± arccos(1/4) + 2πn, so θ = ± (1/2) arccos(1/4) + πn.Then, compute y at θ = (1/2) arccos(1/4).Let me denote φ = arccos(1/4). So, cosφ = 1/4, sinφ = √(1 - 1/16) = √(15/16) = √15 / 4.Then, θ = φ/2.Compute y = sin3θ sinθ.First, compute sinθ and sin3θ.Using half-angle identities:sinθ = sin(φ/2) = √[(1 - cosφ)/2] = √[(1 - 1/4)/2] = √[(3/4)/2] = √(3/8) = √6 / 4.Similarly, cosθ = cos(φ/2) = √[(1 + cosφ)/2] = √[(1 + 1/4)/2] = √[(5/4)/2] = √(5/8) = √10 / 4.Now, compute sin3θ. Using the identity sin3θ = 3sinθ - 4sin³θ.So, sin3θ = 3*(√6 / 4) - 4*(√6 / 4)^3.Compute each term:3*(√6 / 4) = 3√6 / 4.(√6 / 4)^3 = (6√6) / 64 = (3√6) / 32.So, 4*(√6 / 4)^3 = 4*(3√6 / 32) = 12√6 / 32 = 3√6 / 8.Thus, sin3θ = 3√6 / 4 - 3√6 / 8 = (6√6 / 8 - 3√6 / 8) = 3√6 / 8.Now, y = sin3θ sinθ = (3√6 / 8) * (√6 / 4) = (3*6) / (32) = 18 / 32 = 9 / 16.So, y = 9/16.Wait, but according to the user's method, it's 4√3 / 9 ≈ 0.7698, while 9/16 ≈ 0.5625.So, which one is correct?Wait, maybe the user made a mistake in their expansion. Let me check their steps.The user wrote:y = sin3θ sinθ = (sin2θ cosθ + cos2θ sinθ) sinθ.Wait, that's using the identity sin(A + B) = sinA cosB + cosA sinB. So, sin3θ = sin(2θ + θ) = sin2θ cosθ + cos2θ sinθ.Then, y = sin3θ sinθ = (sin2θ cosθ + cos2θ sinθ) sinθ.Expanding this, they got:= sin2θ cosθ sinθ + cos2θ sin²θ.Then, they expressed sin2θ as 2 sinθ cosθ:= 2 sinθ cosθ * cosθ sinθ + cos2θ sin²θ.Wait, that seems incorrect. Because sin2θ cosθ sinθ = 2 sinθ cosθ * cosθ sinθ = 2 sin²θ cos²θ.Wait, but in the user's step, they wrote:= (2 sinθ cosθ cosθ + cos2θ sin²θ) = 2 sinθ cos²θ + cos2θ sin²θ.Wait, that's correct. Then, they expressed cos²θ as 1 - sin²θ:= 2 sinθ (1 - sin²θ) + (1 - 2 sin²θ) sin²θ.So, expanding:= 2 sinθ - 2 sin³θ + sin²θ - 2 sin⁴θ.Wait, but in the user's step, they wrote:= 2 sinθ - 3 sin³θ + sin³θ = 2 sinθ - 2 sin³θ.Wait, that seems incorrect. Because 2 sinθ - 2 sin³θ + sin²θ - 2 sin⁴θ.Wait, the user seems to have combined terms incorrectly.Wait, let me re-express:After expanding, y = 2 sinθ (1 - sin²θ) + (1 - 2 sin²θ) sin²θ.= 2 sinθ - 2 sin³θ + sin²θ - 2 sin⁴θ.So, combining like terms:= 2 sinθ + sin²θ - 2 sin³θ - 2 sin⁴θ.Wait, that's different from what the user wrote. The user wrote:= 2 sinθ - 3 sin³θ + sin³θ = 2 sinθ - 2 sin³θ.Which skips the sin²θ term and incorrectly combines the sin³θ terms.So, the user made a mistake in their algebra when expanding. They incorrectly combined terms, leading to an incorrect expression for y.Therefore, their subsequent steps leading to y = 2u - 2u³ are incorrect because they missed the sin²θ term.So, the correct expression for y is y = 2 sinθ + sin²θ - 2 sin³θ - 2 sin⁴θ.But that complicates things. Alternatively, using the product-to-sum formula is more straightforward and led us to y = 9/16.Therefore, the correct maximum y-coordinate is 9/16.Wait, but let me confirm with another approach. Let me parametrize θ and compute y numerically.Let me choose θ such that sinθ = √6 / 4 ≈ 0.6124.So, θ ≈ arcsin(0.6124) ≈ 0.661 radians ≈ 37.9 degrees.Compute y = sin(3θ) sinθ.First, compute 3θ ≈ 1.983 radians ≈ 113.7 degrees.Compute sin(3θ) ≈ sin(1.983) ≈ 0.917.Compute sinθ ≈ 0.6124.So, y ≈ 0.917 * 0.6124 ≈ 0.562.Which is approximately 9/16 = 0.5625. So, that matches.On the other hand, if I take θ where sinθ = √3 / 3 ≈ 0.577.Then, θ ≈ arcsin(0.577) ≈ 0.615 radians ≈ 35.26 degrees.Compute 3θ ≈ 1.845 radians ≈ 105.7 degrees.Compute sin(3θ) ≈ sin(1.845) ≈ 0.961.Compute sinθ ≈ 0.577.So, y ≈ 0.961 * 0.577 ≈ 0.554.Which is less than 0.5625.Wait, so the maximum y is indeed 9/16, not 4√3 / 9.Therefore, the user's initial approach had an algebraic mistake, leading to an incorrect maximum value.So, the correct maximum y-coordinate is 9/16.

💡Okay, so I have this problem: "How many ways are there to put 6 balls into 4 boxes if the balls are indistinguishable but the boxes are?" Hmm, let me try to understand what this means.First, the balls are indistinguishable, which means I can't tell them apart. So, it doesn't matter which specific ball goes into which box; what matters is how many balls are in each box. On the other hand, the boxes are distinguishable, meaning each box is unique. Maybe they're different colors or labeled differently, so putting balls into Box 1 versus Box 2 is a different arrangement.So, essentially, I need to find the number of ways to distribute 6 identical balls into 4 distinct boxes. This sounds like a combinatorics problem, specifically something related to combinations with repetition or partitions.I remember that when dealing with indistinct objects and distinct containers, the formula for combinations with repetition is useful. The formula is C(n + k - 1, k - 1), where n is the number of objects and k is the number of containers. In this case, n is 6 balls and k is 4 boxes.Let me write that down:Number of ways = C(6 + 4 - 1, 4 - 1) = C(9, 3)Now, I need to compute C(9, 3). The combination formula is C(n, r) = n! / (r! * (n - r)!).So, plugging in the numbers:C(9, 3) = 9! / (3! * (9 - 3)!) = 9! / (3! * 6!) Calculating factorials:9! = 362880 3! = 6 6! = 720So,C(9, 3) = 362880 / (6 * 720) = 362880 / 4320Now, dividing 362880 by 4320:First, let's see how many times 4320 goes into 362880.Dividing both numerator and denominator by 10: 36288 / 432Divide numerator and denominator by 48: 756 / 9 = 84So, C(9, 3) = 84Therefore, there are 84 ways to distribute 6 indistinguishable balls into 4 distinguishable boxes.Wait, just to make sure I didn't make a mistake, let me think of another way to approach this problem. Maybe by enumerating all possible distributions.So, we can think of this as finding the number of non-negative integer solutions to the equation:x1 + x2 + x3 + x4 = 6Where each xi represents the number of balls in box i.This is a classic stars and bars problem. The formula for the number of solutions is indeed C(n + k - 1, k - 1), which in this case is C(6 + 4 - 1, 4 - 1) = C(9, 3) = 84.Alternatively, if I were to list out all possible distributions, it would take a lot of time, but for smaller numbers, it's manageable. For example, distributions like (6,0,0,0), (5,1,0,0), (4,2,0,0), etc., and counting the permutations for each case. But since the boxes are distinguishable, each unique arrangement counts separately.But since the formula gives 84, and my alternative reasoning aligns with it, I feel confident that 84 is the correct answer.Another way to think about it is using generating functions. The generating function for each box is (1 + x + x^2 + x^3 + x^4 + x^5 + x^6), since each box can hold 0 to 6 balls. Since there are 4 boxes, the generating function would be (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)^4. The coefficient of x^6 in this expansion would give the number of ways. However, calculating this manually would be tedious, but it's another method to confirm the result.Alternatively, I can recall that the number of non-negative integer solutions to x1 + x2 + x3 + x4 = 6 is equal to C(6 + 4 - 1, 4 - 1) = C(9,3) = 84, which matches my earlier calculation.So, after considering multiple approaches—combinations with repetition, stars and bars, and generating functions—I consistently arrive at the same answer: 84.I think I'm confident enough that 84 is the correct number of ways to distribute 6 indistinguishable balls into 4 distinguishable boxes.

💡Okay, so I have this problem where I need to find the number of sets of two non-empty disjoint subsets of the set T, which is {1, 2, 3, ..., 12}. Then, I have to find the remainder when that number is divided by 500. Hmm, let me try to break this down step by step.First, let's understand what is being asked. We need to count the number of ways to choose two subsets A and B from T such that A and B are both non-empty and disjoint. Also, since the problem mentions "sets of two subsets," the order doesn't matter. That is, {A, B} is the same as {B, A}, so we don't want to count them twice.Alright, so how do I approach this? I remember that when dealing with subsets and counting problems, sometimes it's helpful to consider each element and how it can be assigned. Since A and B are disjoint, each element can either be in A, in B, or in neither. That gives us three choices per element.So, if I think about each element in T, it can be in A, in B, or in neither. That means for each of the 12 elements, there are 3 choices. Therefore, the total number of ways to assign all elements is 3^12. But wait, that counts all possible ordered pairs (A, B), including cases where A or B might be empty.But the problem specifies that both A and B must be non-empty. So, I need to subtract the cases where A is empty or B is empty. Let me think about that.If A is empty, then each element can either be in B or in neither. So, that's 2^12 possibilities. Similarly, if B is empty, each element can either be in A or in neither, which is another 2^12 possibilities. But wait, if I subtract both of these, I might be overcounting the case where both A and B are empty. That case was subtracted twice, so I need to add it back once.So, using inclusion-exclusion principle, the number of ordered pairs (A, B) where A and B are disjoint and non-empty is 3^12 - 2*2^12 + 1. Let me write that down:Number of ordered pairs = 3^12 - 2*2^12 + 1.But the problem asks for the number of sets {A, B}, not ordered pairs. Since {A, B} is the same as {B, A}, I need to divide by 2 to account for this duplication. So, the number of unordered pairs is (3^12 - 2*2^12 + 1)/2.Let me compute that. First, let's calculate 3^12. I know that 3^5 is 243, 3^10 is 243^2, which is 59049, and then 3^12 is 59049 * 9 = 531441.Next, 2^12 is 4096. So, 2*2^12 is 8192.So, plugging in the numbers:Number of ordered pairs = 531441 - 8192 + 1 = 531441 - 8192 is 523249, plus 1 is 523250.Then, dividing by 2 gives us 261625.So, m is 261625. Now, the problem asks for m modulo 500. So, I need to compute 261625 divided by 500 and find the remainder.Let me do that. 500 goes into 261625 how many times? Well, 500 * 500 is 250000. So, subtracting that from 261625 gives 11625.Now, 500 goes into 11625 how many times? 500 * 23 is 11500. So, subtracting 11500 from 11625 gives 125.So, the remainder is 125.Wait, let me check that again to make sure I didn't make a mistake. 500 * 523 is 261500. Then, 261625 - 261500 is 125. Yeah, that's correct.So, the remainder when m is divided by 500 is 125.I think that's it. Let me recap to make sure I didn't skip any steps or make any errors. I considered each element having three choices, calculated the total number of ordered pairs, subtracted the cases where A or B is empty, adjusted for overcounting, then divided by 2 to get unordered pairs. Then, I computed the result modulo 500. It all seems to add up.**Final Answer**The remainder when ( m ) is divided by 500 is (boxed{125}).

💡Okay, so I have this problem here: the curve ( x^2 - 4y^2 = 16 ) undergoes a scaling transformation in the direction of the y-axis with a scaling factor of ( k = 2 ). I need to find the equation of the curve after this transformation.First, I need to recall what a scaling transformation is. From what I remember, scaling a curve means stretching or shrinking it along a particular axis. In this case, it's along the y-axis. The scaling factor ( k = 2 ) tells me that the curve will be stretched vertically by a factor of 2.Now, how do I apply this scaling to the equation of the curve? I think it involves substituting a scaled version of y into the original equation. Since we're scaling along the y-axis, we replace ( y ) with ( frac{y}{k} ) in the equation. Let me write that down.Original equation: ( x^2 - 4y^2 = 16 ).After scaling, replace ( y ) with ( frac{y}{2} ):( x^2 - 4left(frac{y}{2}right)^2 = 16 ).Now, let me simplify this equation step by step. First, calculate ( left(frac{y}{2}right)^2 ):( left(frac{y}{2}right)^2 = frac{y^2}{4} ).Substitute that back into the equation:( x^2 - 4 times frac{y^2}{4} = 16 ).Simplify the multiplication:( x^2 - frac{4y^2}{4} = 16 ).The 4 in the numerator and denominator cancel out:( x^2 - y^2 = 16 ).So, after the scaling transformation, the equation of the curve becomes ( x^2 - y^2 = 16 ).Wait, let me double-check that. If I scale y by 2, does that mean stretching it vertically? Yes, because scaling by a factor greater than 1 stretches the graph. So, replacing ( y ) with ( frac{y}{2} ) effectively stretches the graph vertically by a factor of 2. That makes sense because if I think about a point (x, y) on the original curve, after scaling, it would be at (x, 2y), which is further away from the x-axis, hence stretching.Let me also consider what the original curve looks like. The equation ( x^2 - 4y^2 = 16 ) is a hyperbola centered at the origin, opening to the left and right along the x-axis. The standard form of a hyperbola is ( frac{x^2}{a^2} - frac{y^2}{b^2} = 1 ). So, if I divide both sides by 16, I get:( frac{x^2}{16} - frac{y^2}{4} = 1 ).Here, ( a^2 = 16 ) so ( a = 4 ), and ( b^2 = 4 ) so ( b = 2 ). The vertices are at ( (pm 4, 0) ) and the foci are at ( (pm sqrt{a^2 + b^2}, 0) = (pm sqrt{16 + 4}, 0) = (pm sqrt{20}, 0) ).After scaling, the equation becomes ( x^2 - y^2 = 16 ). Let me write that in standard form by dividing both sides by 16:( frac{x^2}{16} - frac{y^2}{16} = 1 ).So now, both ( a^2 ) and ( b^2 ) are 16, meaning ( a = 4 ) and ( b = 4 ). The vertices are still at ( (pm 4, 0) ), but the foci are now at ( (pm sqrt{a^2 + b^2}, 0) = (pm sqrt{16 + 16}, 0) = (pm sqrt{32}, 0) ).This makes sense because scaling the y-axis by a factor of 2 would stretch the hyperbola vertically, making the branches steeper. The asymptotes of the original hyperbola were ( y = pm frac{b}{a}x = pm frac{2}{4}x = pm frac{1}{2}x ). After scaling, the asymptotes become ( y = pm frac{b}{a}x = pm frac{4}{4}x = pm x ), which are indeed steeper, confirming that the scaling has affected the y-axis.Let me also think about how points transform. Suppose I have a point (x, y) on the original curve. After scaling, the point becomes (x, 2y). So, if I plug (x, 2y) into the original equation, it should satisfy the transformed equation.Let's test this. Take a point on the original curve, say (4, 0). Plugging into the original equation: ( 4^2 - 4(0)^2 = 16 - 0 = 16 ), which satisfies the equation. After scaling, the point becomes (4, 0) since y is 0. Plugging into the transformed equation: ( 4^2 - 0^2 = 16 - 0 = 16 ), which also satisfies the transformed equation.Another point: Let's say (0, y). On the original curve, plugging x=0: ( 0 - 4y^2 = 16 ) leads to ( y^2 = -4 ), which is not possible, so there are no points on the original curve where x=0. That makes sense because it's a hyperbola opening along the x-axis.After scaling, plugging x=0 into the transformed equation: ( 0 - y^2 = 16 ) leads to ( y^2 = -16 ), still no real solutions, which is consistent.Let me try another point. Suppose I take a point where y is non-zero. Let's solve for y in the original equation when x=6. Then:( 6^2 - 4y^2 = 16 )( 36 - 4y^2 = 16 )( -4y^2 = 16 - 36 )( -4y^2 = -20 )( y^2 = 5 )( y = pm sqrt{5} )So, the points are (6, ( sqrt{5} )) and (6, ( -sqrt{5} )).After scaling, these points become (6, ( 2sqrt{5} )) and (6, ( -2sqrt{5} )). Let's plug into the transformed equation:( 6^2 - (2sqrt{5})^2 = 36 - 4*5 = 36 - 20 = 16 ), which satisfies the equation. Similarly for the negative y.So, this seems consistent. The scaling transformation in the y-axis direction with k=2 affects the y-coordinates by multiplying them by 2, which in turn affects the equation by replacing y with y/2 to maintain the equality.I think I've covered the steps thoroughly. I applied the scaling transformation correctly by substituting y with y/k, simplified the equation, and verified with specific points to ensure the transformation is accurate. The final equation after scaling is indeed ( x^2 - y^2 = 16 ).

💡Okay, so I have this problem about a function f(x) = 2 sin(ωx - π/6) + 1. They mention that the distance between two adjacent axes of symmetry of its graph is π/2. I need to find the analytical expression of f(x) and then find its maximum value when x is between 0 and π/2.First, let me recall what the graph of a sine function looks like. The general form is A sin(Bx + C) + D. In this case, A is 2, which is the amplitude, so the graph will oscillate between 2 and -2, but then it's shifted up by 1, so it'll oscillate between 3 and -1. The ω is the angular frequency, which affects the period of the sine wave. The phase shift is π/6, which shifts the graph horizontally.Now, the problem mentions the distance between two adjacent axes of symmetry. Hmm, axes of symmetry for a sine wave... I think that refers to the vertical lines that pass through the peaks or troughs of the sine wave. For a standard sine wave, these would be at the midpoints between the peaks and troughs, right? So, for the standard sin(x), the axes of symmetry would be at x = π/2, 3π/2, etc., which are the points where the sine function reaches its maximum and minimum.Wait, actually, for a sine wave, the axes of symmetry are the lines that pass through the peaks and troughs. So, for sin(x), the peaks are at π/2, 5π/2, etc., and the troughs are at 3π/2, 7π/2, etc. So, the distance between two adjacent axes of symmetry would be half the period, right? Because between a peak and the next trough is half a period, and that's the distance between two adjacent symmetry axes.So, if the distance between two adjacent axes of symmetry is π/2, that means half the period is π/2. Therefore, the full period T would be π. Because half of T is π/2, so T = π.Now, the period T of a sine function is given by T = 2π / ω. So, if T = π, then:π = 2π / ωSolving for ω:Multiply both sides by ω: π * ω = 2πDivide both sides by π: ω = 2So, ω is 2. Therefore, the function becomes f(x) = 2 sin(2x - π/6) + 1.Alright, that should answer part (1). Now, moving on to part (2), where I need to find the maximum value of f(x) when x is in [0, π/2].So, f(x) = 2 sin(2x - π/6) + 1. To find its maximum, I need to find the maximum value of sin(2x - π/6) in the interval x ∈ [0, π/2].First, let's find the range of the argument inside the sine function. Let θ = 2x - π/6. When x = 0, θ = -π/6. When x = π/2, θ = 2*(π/2) - π/6 = π - π/6 = 5π/6.So, θ ranges from -π/6 to 5π/6. Now, the sine function reaches its maximum value of 1 at π/2. So, I need to check if π/2 is within the interval [-π/6, 5π/6]. Yes, π/2 is approximately 1.57, and 5π/6 is approximately 2.618, so π/2 is within that interval.Therefore, the maximum value of sin(θ) is 1, which occurs when θ = π/2. So, substituting back, sin(2x - π/6) = 1 when 2x - π/6 = π/2.Solving for x:2x - π/6 = π/2Add π/6 to both sides:2x = π/2 + π/6 = (3π/6 + π/6) = 4π/6 = 2π/3Divide both sides by 2:x = π/3So, at x = π/3, which is within [0, π/2], the sine function reaches its maximum value of 1. Therefore, f(x) at this point is:f(π/3) = 2*1 + 1 = 3Therefore, the maximum value of f(x) in the interval [0, π/2] is 3.Wait, let me double-check. Is there any other point where sin(θ) could be higher? Since θ goes from -π/6 to 5π/6, and the sine function reaches its maximum at π/2, which is within that interval, so yes, 1 is the maximum value. So, f(x) = 2*1 + 1 = 3 is indeed the maximum.I think that's it. So, part (1) gives us f(x) = 2 sin(2x - π/6) + 1, and part (2) gives the maximum value as 3.**Final Answer**(1) The analytical expression of the function is boxed{2sinleft(2x - frac{pi}{6}right) + 1}.(2) The maximum value of ( f(x) ) when ( x in [0, frac{pi}{2}] ) is boxed{3}.

💡Alright, so I have this problem here about three-digit numbers of the form ( x y 0 ), where ( x ) and ( y ) are digits, and ( x ) can't be zero. I need to find out how many of these numbers are divisible by both 11 and 20. Hmm, okay, let's break this down step by step.First off, let's understand what the number ( x y 0 ) represents. Since it's a three-digit number, ( x ) is the hundreds digit, ( y ) is the tens digit, and the units digit is 0. So, the number can be written as ( 100x + 10y + 0 ), which simplifies to ( 100x + 10y ).Now, the problem is asking for numbers that are divisible by both 11 and 20. I remember that if a number is divisible by both 11 and 20, it must be divisible by their least common multiple (LCM). Let me calculate that real quick. The LCM of 11 and 20 is 220 because 11 is a prime number and 20 is ( 2^2 times 5 ), so there are no common factors. Therefore, the LCM is just ( 11 times 20 = 220 ).So, essentially, I need to find all three-digit numbers of the form ( x y 0 ) that are divisible by 220. Let's think about how to approach this.First, let's recall the divisibility rules for 11 and 20 to see if we can apply them here.**Divisibility by 11:**A number is divisible by 11 if the difference between the sum of its digits in the odd positions and the sum of its digits in the even positions is a multiple of 11 (including zero). For the number ( x y 0 ), the digits are in positions 1 (hundreds), 2 (tens), and 3 (units). So, the sum of digits in odd positions is ( x + 0 = x ), and the sum of digits in even positions is ( y ). Therefore, the difference is ( x - y ). For the number to be divisible by 11, ( x - y ) must be a multiple of 11, which includes zero.Since ( x ) and ( y ) are single digits (0-9), the only way ( x - y ) can be a multiple of 11 is if ( x = y ). Because if ( x ) and ( y ) are both single digits, the maximum difference is 9 (if ( x = 9 ) and ( y = 0 )), which is less than 11. So, ( x - y = 0 ) implies ( x = y ).**Divisibility by 20:**A number is divisible by 20 if it is divisible by both 4 and 5. Divisibility by 5 is straightforward: the number must end with 0 or 5. In our case, the number ends with 0, so it's already divisible by 5. Now, for divisibility by 4, the rule is that the last two digits of the number must form a number that is divisible by 4. In our case, the last two digits are ( y 0 ), which is ( 10y ).So, ( 10y ) must be divisible by 4. Let's find out for which values of ( y ) this is true.( 10y ) divisible by 4 means ( 10y mod 4 = 0 ). Let's compute ( 10y ) for ( y = 0 ) to ( y = 9 ):- ( y = 0 ): ( 10 times 0 = 0 ) → 0 is divisible by 4.- ( y = 1 ): ( 10 times 1 = 10 ) → 10 ÷ 4 = 2.5 → Not divisible.- ( y = 2 ): ( 10 times 2 = 20 ) → 20 ÷ 4 = 5 → Divisible.- ( y = 3 ): ( 10 times 3 = 30 ) → 30 ÷ 4 = 7.5 → Not divisible.- ( y = 4 ): ( 10 times 4 = 40 ) → 40 ÷ 4 = 10 → Divisible.- ( y = 5 ): ( 10 times 5 = 50 ) → 50 ÷ 4 = 12.5 → Not divisible.- ( y = 6 ): ( 10 times 6 = 60 ) → 60 ÷ 4 = 15 → Divisible.- ( y = 7 ): ( 10 times 7 = 70 ) → 70 ÷ 4 = 17.5 → Not divisible.- ( y = 8 ): ( 10 times 8 = 80 ) → 80 ÷ 4 = 20 → Divisible.- ( y = 9 ): ( 10 times 9 = 90 ) → 90 ÷ 4 = 22.5 → Not divisible.So, the values of ( y ) that make ( 10y ) divisible by 4 are ( y = 0, 2, 4, 6, 8 ).But wait, earlier we concluded that ( x = y ) for divisibility by 11. So, ( x ) must equal ( y ), and ( x ) cannot be zero. Therefore, ( y ) cannot be zero either because ( x ) can't be zero. So, from the list above, ( y ) can be 2, 4, 6, or 8.Therefore, the possible values for ( x ) and ( y ) are:- ( x = y = 2 )- ( x = y = 4 )- ( x = y = 6 )- ( x = y = 8 )Thus, the numbers are:- 220- 440- 660- 880Let me verify each of these numbers to ensure they are indeed divisible by both 11 and 20.**Verification:**1. **220:** - Divisible by 11: ( 2 - 2 = 0 ), which is a multiple of 11. - Divisible by 20: Ends with 20, which is divisible by 20. 2. **440:** - Divisible by 11: ( 4 - 4 = 0 ), which is a multiple of 11. - Divisible by 20: Ends with 40, which is divisible by 20. 3. **660:** - Divisible by 11: ( 6 - 6 = 0 ), which is a multiple of 11. - Divisible by 20: Ends with 60, which is divisible by 20. 4. **880:** - Divisible by 11: ( 8 - 8 = 0 ), which is a multiple of 11. - Divisible by 20: Ends with 80, which is divisible by 20.All four numbers satisfy both divisibility conditions.So, in conclusion, there are 4 three-digit numbers of the form ( x y 0 ) that are divisible by both 11 and 20.