💡First, I need to determine the total number of ways to select two distinct numbers from the set ({1, 2, 3, 4, 5, 6}). This can be calculated using the combination formula:[text{Total ways} = binom{6}{2} = 15]Next, I'll identify the conditions under which the sum of two numbers is even. An even sum occurs in two scenarios:1. Both numbers are odd.2. Both numbers are even.Looking at the set, there are 3 odd numbers (1, 3, 5) and 3 even numbers (2, 4, 6).Calculating the number of ways to select two odd numbers:[text{Ways to select two odd numbers} = binom{3}{2} = 3]Similarly, calculating the number of ways to select two even numbers:[text{Ways to select two even numbers} = binom{3}{2} = 3]Adding these, the total number of successful outcomes is:[3 + 3 = 6]Finally, the probability of selecting two numbers with an even sum is:[text{Probability} = frac{6}{15} = frac{2}{5}]

💡Okay, so I have this problem here that asks which of the given equations does NOT have a solution. There are five options labeled (A) through (E). Let me go through each one step by step to figure out which one doesn't have a solution.Starting with option (A): ((x - 5)^2 = 0). Hmm, squaring any real number will always give a non-negative result, right? So, if ((x - 5)^2 = 0), that means (x - 5) must be zero because the square of zero is zero. Therefore, solving for (x) gives (x = 5). Okay, so equation (A) has a solution at (x = 5). That means (A) is not the answer we're looking for.Moving on to option (B): (|x| + 7 = 0). Absolute value, (|x|), is always non-negative. So, (|x|) is zero or a positive number. If I add 7 to it, the smallest value this expression can take is 7 (when (x = 0)). Therefore, (|x| + 7) is always greater than or equal to 7, which means it can never be zero. So, this equation doesn't have any real solutions. Hmm, this looks like a strong candidate for the answer, but I should check the other options just to be sure.Option (C): (sqrt{x + 9} - 3 = 0). Let's solve this. First, I can rewrite it as (sqrt{x + 9} = 3). To get rid of the square root, I'll square both sides, which gives me (x + 9 = 9). Subtracting 9 from both sides, I get (x = 0). Now, I should check if this solution is valid. The expression inside the square root, (x + 9), must be non-negative. Plugging (x = 0) in, we get (0 + 9 = 9), which is positive. So, (x = 0) is a valid solution. Therefore, equation (C) has a solution, so it's not the answer.Next is option (D): (sqrt[3]{x + 4} - 1 = 0). Let's solve this. Adding 1 to both sides gives (sqrt[3]{x + 4} = 1). To eliminate the cube root, I'll cube both sides, resulting in (x + 4 = 1). Subtracting 4 from both sides, I find (x = -3). Cube roots are defined for all real numbers, so there's no issue with the domain here. Thus, equation (D) has a solution at (x = -3), so it's not the answer we're looking for.Finally, option (E): (|x + 6| - 5 = 0). Let's solve this. Adding 5 to both sides gives (|x + 6| = 5). The absolute value equation (|x + 6| = 5) has two solutions: (x + 6 = 5) and (x + 6 = -5). Solving the first equation, (x + 6 = 5), gives (x = -1). Solving the second equation, (x + 6 = -5), gives (x = -11). Both solutions are real numbers, so equation (E) has solutions at (x = -1) and (x = -11). Therefore, (E) is not the answer.After going through all the options, it seems that option (B) is the only equation that doesn't have a solution because the sum of an absolute value and 7 can never be zero. All the other equations have at least one real solution. So, I'm confident that (B) is the correct answer.**Final Answer**boxed{B}

💡First, I need to translate the problem into a mathematical equation. The problem states that a certain number multiplied by 7, minus 6, minus 12, equals the number multiplied by 4. Let's denote the certain number as ( x ).So, the equation becomes:[ 7x - 6 - 12 = 4x ]Next, I'll simplify the left side of the equation by combining the constant terms:[ 7x - 18 = 4x ]To isolate ( x ), I'll subtract ( 4x ) from both sides:[ 3x - 18 = 0 ]Then, I'll add 18 to both sides to get:[ 3x = 18 ]Finally, I'll divide both sides by 3 to solve for ( x ):[ x = 6 ]

💡Okay, so I have this problem about Bernoulli trials and probability. Let me try to understand what it's asking. There are 7 Bernoulli trials, and a success is defined as a hit that topples one of the two posts attached to a red fence section. The probability of success in each trial is 2/7. I need to find the probability that the red section does not suffer any damage, meaning there are zero successes in these 7 trials.Alright, so first, let me recall what a Bernoulli trial is. It's an experiment with exactly two possible outcomes: success or failure. In this case, success is hitting one of the two posts, and failure is not hitting them. The probability of success is given as 2/7, which makes sense because there are 7 posts in total, and 2 of them are relevant to the red section.Now, since we're dealing with multiple Bernoulli trials, this sounds like a binomial distribution problem. The binomial distribution gives the probability of having exactly k successes in n independent trials, with the probability of success in a single trial being p. The formula for the binomial distribution is:P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)Where C(n, k) is the combination of n things taken k at a time.In this problem, we're interested in the probability of 0 successes (k = 0) out of 7 trials (n = 7), with the probability of success in each trial being 2/7 (p = 2/7).So, plugging these values into the formula:P(X = 0) = C(7, 0) * (2/7)^0 * (1 - 2/7)^(7 - 0)Let me compute each part step by step.First, C(7, 0) is the number of ways to choose 0 successes out of 7 trials. I remember that C(n, 0) is always 1 because there's only one way to choose nothing.Next, (2/7)^0 is any number to the power of 0, which is 1.Then, (1 - 2/7) is 5/7. So, (5/7)^7 is the probability of failing all 7 trials.Putting it all together:P(X = 0) = 1 * 1 * (5/7)^7Now, I need to calculate (5/7)^7. Let me compute that.First, 5 divided by 7 is approximately 0.714285714.Raising this to the 7th power:0.714285714^7Let me compute this step by step.0.714285714^2 = 0.714285714 * 0.714285714 ≈ 0.51020408160.5102040816^2 = 0.5102040816 * 0.5102040816 ≈ 0.26040816320.2604081632 * 0.714285714 ≈ 0.18627450980.1862745098 * 0.714285714 ≈ 0.13314285710.1331428571 * 0.714285714 ≈ 0.0952380952So, approximately, (5/7)^7 ≈ 0.0952380952Therefore, the probability of having 0 successes in 7 trials is approximately 0.0952, or 9.52%.Wait, let me double-check my calculations to make sure I didn't make a mistake.First, 5/7 is approximately 0.714285714.0.714285714^2 = 0.51020408160.5102040816^2 = 0.26040816320.2604081632 * 0.714285714 ≈ 0.18627450980.1862745098 * 0.714285714 ≈ 0.13314285710.1331428571 * 0.714285714 ≈ 0.0952380952Yes, that seems consistent. So, (5/7)^7 ≈ 0.0952380952Therefore, the probability that the red section does not suffer any damage is approximately 0.0952, or 9.52%.I think that's the answer. But just to be thorough, let me consider if there's another way to approach this problem.Alternatively, since each trial is independent, the probability of failing all 7 trials is simply (1 - p)^n, where p is the probability of success and n is the number of trials.So, in this case, that would be (5/7)^7, which is exactly what I calculated earlier.Therefore, the probability is indeed approximately 0.0952.I think that's correct. I don't see any mistakes in my reasoning or calculations.

💡Okay, so I've got this problem about mixing solutions with different concentrations of salt and sugar. Let me try to understand it step by step. First, there's a container with a solution that's 10% salt by weight. Then, one-third of this solution is replaced with another solution that's 5% salt and 20% sugar by weight. After that, the mixture is combined with an equal volume of a third solution that's 15% salt by weight but has no sugar. I need to find the salt concentration in the final mixture.Hmm, okay. Let's break it down. Maybe I should start by assuming a certain volume for the original solution to make the calculations easier. Let's say the original solution has a volume of V liters. Since it's 10% salt by weight, the amount of salt in it is 0.10V.Now, one-third of this solution is replaced. So, the volume removed is V/3 liters. The amount of salt removed would be 10% of V/3, which is (10/100)*(V/3) = V/30. So, the remaining salt in the container after removing V/3 liters is 0.10V - V/30.Then, we add the second solution, which is V/3 liters with 5% salt. The amount of salt added is (5/100)*(V/3) = V/60. So, the total salt after this replacement is (0.10V - V/30) + V/60.Let me calculate that: 0.10V is 3V/30, so 3V/30 - V/30 = 2V/30 = V/15. Then, adding V/60 gives V/15 + V/60 = (4V + V)/60 = 5V/60 = V/12. So, after the replacement, the total salt is V/12, and the volume is still V liters because we removed V/3 and added V/3.Now, we add an equal volume of the third solution, which is V liters of 15% salt. The amount of salt in this third solution is 0.15V. So, the total salt in the final mixture is V/12 + 0.15V.Let me convert V/12 to decimal to make it easier: V/12 ≈ 0.0833V. Adding 0.15V gives 0.0833V + 0.15V = 0.2333V.The total volume of the final mixture is V (from the first two solutions) + V (third solution) = 2V liters. So, the salt concentration is 0.2333V / 2V = 0.11665, which is approximately 11.665%.Wait, that seems a bit low. Let me double-check my calculations. Maybe I made a mistake in the fractions.Starting again: Original salt is 0.10V. Removing V/3, which has 0.10*(V/3) = V/30 salt. So, remaining salt is 0.10V - V/30 = (3V/30 - V/30) = 2V/30 = V/15.Adding the second solution: 5% of V/3 is (5/100)*(V/3) = V/60. So, total salt becomes V/15 + V/60 = (4V + V)/60 = 5V/60 = V/12. That's correct.Then, adding the third solution: 15% of V is 0.15V. So, total salt is V/12 + 0.15V. Converting V/12 to decimal: V/12 ≈ 0.0833V. Adding 0.15V gives 0.2333V.Total volume is 2V, so concentration is 0.2333V / 2V = 0.11665, which is 11.665%. Okay, that seems consistent.I think that's the correct answer. So, the salt concentration in the final mixture is approximately 11.665% by weight.

💡Okay, so I have this problem about a 44-gon, which is a polygon with 44 sides. It's called Q₁, and it's drawn on the Cartesian plane. The sum of the x-coordinates of its 44 vertices is 132. Then, they form another 44-gon, Q₂, by taking the midpoints of the sides of Q₁. Then, they do the same thing again to form Q₃ from Q₂. The question is asking for the sum of the x-coordinates of the vertices of Q₃.Hmm, okay. So, I need to figure out how the sum of the x-coordinates changes when we take midpoints of the sides to form a new polygon. Let me think about this step by step.First, let's consider Q₁. It has 44 vertices, each with an x-coordinate. The sum of all these x-coordinates is 132. Let me denote the x-coordinates of Q₁ as x₁, x₂, x₃, ..., x₄₄. So, x₁ + x₂ + x₃ + ... + x₄₄ = 132.Now, Q₂ is formed by taking the midpoints of the sides of Q₁. Each side of Q₁ connects two consecutive vertices, right? So, the midpoint of each side will have coordinates that are the average of the coordinates of the two endpoints. Since we're dealing with x-coordinates, the x-coordinate of each midpoint will be the average of the x-coordinates of the two vertices it connects.So, for example, the first midpoint will be between vertex 1 and vertex 2, so its x-coordinate will be (x₁ + x₂)/2. The next midpoint will be between vertex 2 and vertex 3, so its x-coordinate will be (x₂ + x₃)/2, and so on. Since it's a polygon, the last midpoint will connect vertex 44 back to vertex 1, so its x-coordinate will be (x₄₄ + x₁)/2.Therefore, the x-coordinates of Q₂ are: (x₁ + x₂)/2, (x₂ + x₃)/2, ..., (x₄₄ + x₁)/2.Now, let's compute the sum of these x-coordinates for Q₂. Let me write that out:Sum for Q₂ = [(x₁ + x₂)/2] + [(x₂ + x₃)/2] + ... + [(x₄₄ + x₁)/2]I can factor out the 1/2:Sum for Q₂ = (1/2) * [(x₁ + x₂) + (x₂ + x₃) + ... + (x₄₄ + x₁)]Now, let's look at the terms inside the brackets. Each x-coordinate from Q₁ appears twice, right? For example, x₁ appears in the first term and the last term, x₂ appears in the first and second terms, and so on. So, each x_i for i from 1 to 44 is added twice.Therefore, the sum inside the brackets is 2*(x₁ + x₂ + ... + x₄₄). We know that x₁ + x₂ + ... + x₄₄ is 132, so this becomes 2*132 = 264.So, Sum for Q₂ = (1/2)*264 = 132.Wait, that's interesting. The sum of the x-coordinates for Q₂ is also 132, just like Q₁. So, taking midpoints didn't change the sum of the x-coordinates.Now, the problem says we form Q₃ by taking midpoints of the sides of Q₂. So, I guess the same logic applies here. Let me check.Let me denote the x-coordinates of Q₂ as y₁, y₂, ..., y₄₄. Then, the x-coordinates of Q₃ will be the midpoints of the sides of Q₂, so each x-coordinate of Q₃ will be (y₁ + y₂)/2, (y₂ + y₃)/2, ..., (y₄₄ + y₁)/2.Following the same reasoning as before, the sum of the x-coordinates of Q₃ will be:Sum for Q₃ = [(y₁ + y₂)/2] + [(y₂ + y₃)/2] + ... + [(y₄₄ + y₁)/2]Factor out the 1/2:Sum for Q₃ = (1/2) * [(y₁ + y₂) + (y₂ + y₃) + ... + (y₄₄ + y₁)]Again, each y_i appears twice in the sum inside the brackets, so:Sum inside brackets = 2*(y₁ + y₂ + ... + y₄₄) = 2*132 = 264Therefore, Sum for Q₃ = (1/2)*264 = 132So, the sum of the x-coordinates for Q₃ is also 132.Wait, so every time we take midpoints to form a new polygon, the sum of the x-coordinates remains the same? That seems counterintuitive at first, but when I break it down, it makes sense because each time we're averaging pairs of points, but each original point is contributing to two midpoints, so their total contribution remains the same.Let me think about it another way. If I have a set of points and I take midpoints between consecutive points, the centroid (which is the average of all the points) should remain the same, right? Because the centroid is just the average position, and taking midpoints doesn't move the overall center of mass. So, if the sum of the x-coordinates is related to the centroid, then it makes sense that the sum remains the same.In this case, the sum of the x-coordinates is 132 for Q₁, which means the average x-coordinate is 132/44 = 3. So, the centroid is at x = 3. Then, when we form Q₂ and Q₃, the centroid remains at x = 3, so the sum of the x-coordinates remains 132.Therefore, regardless of how many times we take midpoints to form new polygons, the sum of the x-coordinates will stay the same, as long as we're dealing with polygons with the same number of sides and taking midpoints in the same way.So, in conclusion, the sum of the x-coordinates of the vertices of Q₃ is 132.**Final Answer**boxed{132}