💡First, I need to determine the total number of marbles in the bag. There are 3 red, 3 blue, 2 green, and 2 yellow marbles, which adds up to 10 marbles in total.Next, I want to find out how many ways there are to select 4 marbles out of these 10. This can be calculated using the combination formula, which is:[binom{10}{4} = frac{10!}{4!(10-4)!} = 210]Now, I need to calculate the number of favorable outcomes where one marble of each color (red, blue, green, and yellow) is selected. For each color, I calculate the number of ways to choose one marble:- Red: (binom{3}{1} = 3)- Blue: (binom{3}{1} = 3)- Green: (binom{2}{1} = 2)- Yellow: (binom{2}{1} = 2)Multiplying these together gives the total number of favorable outcomes:[3 times 3 times 2 times 2 = 36]Finally, I calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:[frac{36}{210} = frac{6}{35}]Therefore, the probability of selecting one marble of each color is (frac{6}{35}).

💡First, let's denote the third number as ( X ).The first number is 35% less than ( X ), which means it is 65% of ( X ).The second number is 42% less than ( X ), which means it is 58% of ( X ).To find how much percent the second number is less than the first number, we calculate the difference between the two numbers as a percentage of the first number.The difference is ( 65% - 58% = 7% ) of ( X ).Now, we find what percentage this difference is of the first number:[frac{7%}{65%} = frac{7}{65} approx 0.1077]Converting this to a percentage:[0.1077 times 100 approx 10.77%]Therefore, the second number is approximately 10.77% less than the first number.

💡Okay, so I have this problem here: In the expansion of ((x - frac{2}{x})^8), the coefficient of (x^2) is... and then there are four options. Hmm, let me see. I remember that when expanding expressions like this, the binomial theorem is useful. The binomial theorem says that ((a + b)^n = sum_{k=0}^{n} C(n, k) cdot a^{n - k} cdot b^k), where (C(n, k)) is the combination of n things taken k at a time.In this case, (a) is (x) and (b) is (-frac{2}{x}), right? So, applying the binomial theorem, each term in the expansion will look like (C(8, k) cdot x^{8 - k} cdot left(-frac{2}{x}right)^k). Let me write that out:(C(8, k) cdot x^{8 - k} cdot left(-frac{2}{x}right)^k)Simplifying that, the (left(-frac{2}{x}right)^k) can be written as ((-2)^k cdot x^{-k}). So, putting it all together, each term becomes:(C(8, k) cdot (-2)^k cdot x^{8 - k - k}) which simplifies to (C(8, k) cdot (-2)^k cdot x^{8 - 2k})Okay, so each term has an exponent of (x) equal to (8 - 2k). I need to find the term where the exponent is 2 because the question is asking for the coefficient of (x^2). So, I set up the equation:(8 - 2k = 2)Let me solve for (k):(8 - 2k = 2)Subtract 8 from both sides:(-2k = 2 - 8)(-2k = -6)Divide both sides by -2:(k = 3)So, the term where (k = 3) will have the exponent (x^2). Now, let's find the coefficient for that term. The coefficient is (C(8, 3) cdot (-2)^3).First, calculate (C(8, 3)). I remember that (C(n, k) = frac{n!}{k!(n - k)!}). So,(C(8, 3) = frac{8!}{3!5!})Calculating that:(8! = 40320)(3! = 6)(5! = 120)So,(C(8, 3) = frac{40320}{6 times 120} = frac{40320}{720} = 56)Okay, so (C(8, 3) = 56). Now, calculate ((-2)^3):((-2)^3 = -8)Multiply those together:(56 times (-8) = -448)Wait, but looking at the options, none of them have -448. Let me check the options again:A: (-C_8^3 times 2^3)B: (C_8^3 times 2^3)C: (-C_8^3)D: (C_8^3)Hmm, so the options are in terms of combinations and powers of 2, not the actual numerical value. So, I think I need to express the coefficient in terms of (C_8^3) and (2^3).From earlier, the coefficient is (C(8, 3) times (-2)^3). So, that can be written as (-C_8^3 times 2^3), because ((-2)^3 = -8 = -2^3). So, that would be option A.But wait, let me double-check my calculations to make sure I didn't make a mistake. So, (C(8, 3)) is 56, and ((-2)^3) is -8. Multiplying them gives -448. If I use option A: (-C_8^3 times 2^3), that would be (-56 times 8 = -448), which matches. So, that seems correct.Just to be thorough, let me check what the other options would evaluate to:Option B: (C_8^3 times 2^3 = 56 times 8 = 448), which is positive, but our coefficient is negative, so B is incorrect.Option C: (-C_8^3 = -56), which is not the same as -448, so C is incorrect.Option D: (C_8^3 = 56), which is positive and not equal to -448, so D is incorrect.Therefore, the correct answer is A: (-C_8^3 times 2^3).Wait, just to make sure I didn't mix up the signs anywhere. The term is ((x - frac{2}{x})^8), so (b = -frac{2}{x}). So, in the binomial expansion, each term is (C(8, k) cdot x^{8 - k} cdot left(-frac{2}{x}right)^k). So, that's (C(8, k) cdot (-2)^k cdot x^{8 - 2k}). So, when (k = 3), it's (C(8, 3) cdot (-2)^3 cdot x^{2}), which is indeed (-C_8^3 times 2^3). So, yes, A is correct.I think I've covered all the steps and checked my work. I don't see any mistakes in my reasoning.

💡Alright, so I've got this problem here where I need to find the value of (frac{x + y}{x - y}) given that (frac{x}{y} + frac{y}{x} = 8) and that (y > x > 0). Hmm, okay, let's break this down step by step.First off, the equation given is (frac{x}{y} + frac{y}{x} = 8). I remember that when you have fractions like this, it might help to combine them into a single fraction. So, if I find a common denominator, which would be (xy), I can rewrite the equation as:[frac{x^2 + y^2}{xy} = 8]That simplifies to (x^2 + y^2 = 8xy). Okay, so that's a useful equation. Now, the problem asks for (frac{x + y}{x - y}). Hmm, that expression looks a bit tricky, but maybe I can relate it to the equation I just found.Let me denote (a = frac{x + y}{x - y}). My goal is to find the value of (a). Since (y > x > 0), the denominator (x - y) will be negative, and the numerator (x + y) is positive, so (a) should be negative. Wait, but let me double-check that. If (y > x), then (x - y) is negative, and (x + y) is positive, so yes, (a) should be negative. Hmm, but in the solution above, it was concluded that (a) is positive. Maybe I need to be careful here.Anyway, moving on. If I square both sides of the equation (a = frac{x + y}{x - y}), I get:[a^2 = left(frac{x + y}{x - y}right)^2 = frac{(x + y)^2}{(x - y)^2}]Expanding both the numerator and the denominator:[a^2 = frac{x^2 + 2xy + y^2}{x^2 - 2xy + y^2}]Now, from the earlier equation, I know that (x^2 + y^2 = 8xy). So, I can substitute that into the numerator and the denominator:Numerator: (x^2 + y^2 + 2xy = 8xy + 2xy = 10xy)Denominator: (x^2 + y^2 - 2xy = 8xy - 2xy = 6xy)So, substituting back into (a^2):[a^2 = frac{10xy}{6xy} = frac{10}{6} = frac{5}{3}]Therefore, (a = sqrt{frac{5}{3}}) or (a = -sqrt{frac{5}{3}}). But earlier, I thought (a) should be negative because (y > x). Wait, let me think again. If (y > x), then (x - y) is negative, and (x + y) is positive, so (frac{x + y}{x - y}) is negative. So, (a) should be negative. But in the solution above, it was concluded that (a) is positive because (y > x > 0). Hmm, maybe I made a mistake in my initial thought.Wait, let me verify. If (y > x), then (x - y) is negative, and (x + y) is positive, so the ratio is negative. So, (a) should be negative. But in the solution, it was concluded that (a) is positive. That seems contradictory. Maybe I need to check the steps again.Looking back, when I squared (a), I got (a^2 = frac{5}{3}), so (a = pm sqrt{frac{5}{3}}). Since (a) is negative, it should be (-sqrt{frac{5}{3}}). But in the solution, it was concluded that (a) is positive. That must be an error. Hmm, maybe the solution is incorrect.Wait, but let me think again. Maybe I'm misapplying the signs. Let me consider specific numbers to test. Suppose (x = 1) and (y = 2). Then, (frac{x}{y} + frac{y}{x} = frac{1}{2} + 2 = 2.5), which is not 8, but just for testing the sign. Then, (frac{x + y}{x - y} = frac{3}{-1} = -3), which is negative. So, indeed, (a) should be negative. Therefore, the correct answer should be (-sqrt{frac{5}{3}}).But wait, in the original problem, it's stated that (y > x > 0), so (x - y) is negative, and (x + y) is positive, so the ratio is negative. Therefore, the correct value should be negative. So, the solution above might have made a mistake in concluding that (a) is positive.Alternatively, maybe I made a mistake in my reasoning. Let me check the steps again.Starting from:[frac{x}{y} + frac{y}{x} = 8 implies frac{x^2 + y^2}{xy} = 8 implies x^2 + y^2 = 8xy]Then, let (a = frac{x + y}{x - y}). Then,[a^2 = frac{(x + y)^2}{(x - y)^2} = frac{x^2 + 2xy + y^2}{x^2 - 2xy + y^2}]Substituting (x^2 + y^2 = 8xy):[a^2 = frac{8xy + 2xy}{8xy - 2xy} = frac{10xy}{6xy} = frac{5}{3}]So, (a = pm sqrt{frac{5}{3}}). Now, since (y > x), (x - y) is negative, and (x + y) is positive, so (a) is negative. Therefore, (a = -sqrt{frac{5}{3}}).But in the original solution, it was concluded that (a) is positive. That must be incorrect. So, the correct answer should be (-sqrt{frac{5}{3}}).Wait, but let me double-check with actual numbers to make sure. Suppose (x = 1) and (y = phi) (the golden ratio), but that might complicate things. Alternatively, let me solve for (x) and (y) explicitly.From (x^2 + y^2 = 8xy), we can write:[x^2 - 8xy + y^2 = 0]Let me treat this as a quadratic in (x):[x^2 - 8y x + y^2 = 0]Using the quadratic formula:[x = frac{8y pm sqrt{(8y)^2 - 4 cdot 1 cdot y^2}}{2} = frac{8y pm sqrt{64y^2 - 4y^2}}{2} = frac{8y pm sqrt{60y^2}}{2} = frac{8y pm 2ysqrt{15}}{2} = 4y pm ysqrt{15}]Since (x < y), we take the smaller root:[x = 4y - ysqrt{15} = y(4 - sqrt{15})]So, (x = y(4 - sqrt{15})). Now, let's compute (frac{x + y}{x - y}):[frac{x + y}{x - y} = frac{y(4 - sqrt{15}) + y}{y(4 - sqrt{15}) - y} = frac{y(5 - sqrt{15})}{y(3 - sqrt{15})} = frac{5 - sqrt{15}}{3 - sqrt{15}}]Simplify this expression by rationalizing the denominator:Multiply numerator and denominator by (3 + sqrt{15}):[frac{(5 - sqrt{15})(3 + sqrt{15})}{(3 - sqrt{15})(3 + sqrt{15})} = frac{15 + 5sqrt{15} - 3sqrt{15} - 15}{9 - 15} = frac{2sqrt{15}}{-6} = -frac{sqrt{15}}{3} = -sqrt{frac{5}{3}}]Yes, so this confirms that (a = -sqrt{frac{5}{3}}). Therefore, the original solution was incorrect in stating that (a) is positive. The correct value is negative.So, to summarize, the steps are:1. Start with (frac{x}{y} + frac{y}{x} = 8), which simplifies to (x^2 + y^2 = 8xy).2. Let (a = frac{x + y}{x - y}), then square both sides to get (a^2 = frac{10xy}{6xy} = frac{5}{3}).3. Since (y > x), (a) must be negative, so (a = -sqrt{frac{5}{3}}).4. Verified by solving for (x) and (y) explicitly and computing the ratio, which also gives (-sqrt{frac{5}{3}}).Therefore, the correct answer is (-sqrt{frac{5}{3}}).

💡First, I need to calculate the logarithm of the maximum amplitude recorded by the seismograph, which is 1000. Using the logarithm function, I find that (lg 1000 = 3).Next, I calculate the logarithm of the amplitude of the standard earthquake, which is 0.001. Applying the logarithm function again, I determine that (lg 0.001 = -3).Finally, I subtract the logarithm of the standard earthquake's amplitude from the logarithm of the maximum amplitude to find the Richter magnitude: (M = 3 - (-3) = 6).

💡Alright, so I have this problem here. Let me read it again to make sure I understand it properly."If an integer ( n > 9 ) is a solution of the equation ( x^2 - ax + b = 0 ) and the representation of ( a ) in the base-( n ) number system is ( 19 ), then the base-( n ) representation of ( b ) is: - **A)** 19 - **B)** 90 - **C)** 81 - **D)** 99 - **E)** 109"Okay, so we have a quadratic equation ( x^2 - ax + b = 0 ), and one of its solutions is an integer ( n ) which is greater than 9. Also, the number ( a ) when written in base-( n ) is represented as ( 19 ). We need to find the base-( n ) representation of ( b ).Let me break this down step by step.First, I remember that in a quadratic equation of the form ( x^2 - ax + b = 0 ), the sum of the roots is equal to ( a ) and the product of the roots is equal to ( b ). This is known as Vieta's formulas. So, if ( n ) is one of the roots, let's denote the other root as ( m ). Then, according to Vieta's formulas:1. ( n + m = a ) 2. ( n times m = b )So, if I can find the value of ( m ), I can find both ( a ) and ( b ).But wait, the problem says that ( a ) in base-( n ) is ( 19 ). So, I need to convert ( 19 ) from base-( n ) to decimal (base-10) to find the actual value of ( a ).In base-( n ), the number ( 19 ) represents ( 1 times n + 9 times 1 ). So, in decimal, ( a = 1 times n + 9 = n + 9 ).So, ( a = n + 9 ).From Vieta's first formula, ( n + m = a ). Substituting the value of ( a ) we just found:( n + m = n + 9 )If I subtract ( n ) from both sides, I get:( m = 9 )So, the other root ( m ) is 9.Now, using Vieta's second formula, ( n times m = b ). Substituting ( m = 9 ):( b = n times 9 = 9n )So, ( b = 9n ) in decimal.But the question asks for the base-( n ) representation of ( b ). So, I need to convert ( 9n ) from decimal to base-( n ).Let me think about how to do that. When converting a decimal number to another base, we can use division and remainders. But in this case, ( 9n ) is a multiple of ( n ), so it might be straightforward.Let me write ( 9n ) as ( 9 times n + 0 ). So, in base-( n ), this would be represented as two digits: the coefficient of ( n ) is 9, and the constant term is 0. Therefore, ( 9n ) in base-( n ) is written as ( 90 ).Let me double-check that. If I have a number ( 90 ) in base-( n ), that translates to ( 9 times n + 0 times 1 = 9n ) in decimal, which matches our value of ( b ). So, that seems correct.Therefore, the base-( n ) representation of ( b ) is ( 90 ), which corresponds to option **B**.Wait, before I conclude, let me make sure I didn't make any mistakes in my reasoning.1. I used Vieta's formulas correctly: sum of roots is ( a ), product is ( b ). 2. Converted ( 19 ) from base-( n ) to decimal correctly: ( 1 times n + 9 times 1 = n + 9 ). 3. Subtracted ( n ) from both sides to find ( m = 9 ). 4. Calculated ( b = n times 9 = 9n ). 5. Converted ( 9n ) back to base-( n ) as ( 90 ). Everything seems to check out. I don't see any errors in my steps.Just to be thorough, let me consider if there could be another interpretation of the problem. For example, could ( n ) be the base, and ( 19 ) in base-( n ) refer to something else? But no, the problem clearly states that ( a ) is represented as ( 19 ) in base-( n ), so my interpretation is correct.Also, since ( n > 9 ), the digit '9' in base-( n ) is valid because in base-( n ), digits go from 0 to ( n-1 ). So, since ( n > 9 ), '9' is a valid digit in base-( n ).Therefore, I'm confident that my answer is correct.**Final Answer**boxed{B}