💡First, I need to find the repeating decimal representation of the fraction ( frac{5}{13} ). I'll start by performing the division of 5 by 13.1. **Divide 5 by 13:** - 13 goes into 5 zero times. So, I write 0. and proceed with the division. - 13 goes into 50 three times (since ( 13 times 3 = 39 )). Subtracting 39 from 50 leaves a remainder of 11. - Bring down a 0 to make it 110. 13 goes into 110 eight times (since ( 13 times 8 = 104 )). Subtracting 104 from 110 leaves a remainder of 6. - Bring down another 0 to make it 60. 13 goes into 60 four times (since ( 13 times 4 = 52 )). Subtracting 52 from 60 leaves a remainder of 8. - Bring down another 0 to make it 80. 13 goes into 80 six times (since ( 13 times 6 = 78 )). Subtracting 78 from 80 leaves a remainder of 2. - Bring down another 0 to make it 20. 13 goes into 20 one time (since ( 13 times 1 = 13 )). Subtracting 13 from 20 leaves a remainder of 7. - Bring down another 0 to make it 70. 13 goes into 70 five times (since ( 13 times 5 = 65 )). Subtracting 65 from 70 leaves a remainder of 5.2. **Identify the repeating pattern:** - At this point, the remainder is back to 5, which was the original numerator. This indicates that the decimal will start repeating from here.3. **Write the repeating decimal:** - Combining the results of the division, the decimal representation of ( frac{5}{13} ) is ( 0.overline{384615} ).4. **Determine the values of ( c ) and ( d ):** - The repeating part of the decimal is "384615". Since the problem specifies the repeating decimal as ( 0.cdcldots ), it implies that the repeating sequence is two digits long. However, the actual repeating sequence is six digits long. To align with the problem's format, I'll consider the first two digits of the repeating sequence, which are "38".5. **Calculate ( c + d ):** - Here, ( c = 3 ) and ( d = 8 ). - Summing these values gives ( c + d = 3 + 8 = 11 ).

💡Okay, so I have this problem where I need to find the maximum value of |z| given that |z - 6i| + |z - 5| = 7. Hmm, let me try to understand what this means.First, I know that in the complex plane, z can be represented as a point with coordinates (x, y), where z = x + yi. So, |z - 6i| is the distance from z to the point (0, 6), and |z - 5| is the distance from z to the point (5, 0). The equation |z - 6i| + |z - 5| = 7 is saying that the sum of these two distances is 7. Wait a second, that sounds like the definition of an ellipse. An ellipse is the set of all points where the sum of the distances from two fixed points (called foci) is constant. So, in this case, the two foci are at (0, 6) and (5, 0), and the constant sum is 7.But hold on, I remember that for an ellipse, the constant sum has to be greater than the distance between the two foci. Let me check the distance between (0, 6) and (5, 0). Using the distance formula, that's sqrt[(5 - 0)^2 + (0 - 6)^2] = sqrt[25 + 36] = sqrt[61], which is approximately 7.81. Oh, but the sum given is 7, which is less than sqrt(61). That can't be an ellipse because the sum of the distances needs to be greater than the distance between the foci for it to be an ellipse. So, what does this mean? If the sum is less than the distance between the foci, does that mean there are no points z that satisfy this condition? That can't be right because the problem is asking for the maximum |z|, implying that such points do exist.Wait, maybe I made a mistake. Let me think again. If the sum of the distances is equal to the distance between the foci, then the set of points is just the line segment connecting the two foci. So, in this case, since 7 is less than sqrt(61), which is approximately 7.81, it's actually less than the distance between the foci. So, does that mean there are no points z that satisfy |z - 6i| + |z - 5| = 7? That seems contradictory because the problem is asking for the maximum |z|.Hmm, maybe I need to reconsider. Perhaps the problem is correct, and I just need to find the maximum |z| under this condition. Let me try to visualize this. If I have two points, (0, 6) and (5, 0), and I'm looking for points z such that the sum of the distances from z to these two points is 7. Since 7 is less than the distance between the two points, which is sqrt(61) ≈ 7.81, this would imply that the only points that satisfy this condition lie on the line segment connecting (0, 6) and (5, 0). Because if the sum of the distances is equal to the distance between the two points, the point z must lie on the line segment between them.Wait, but in this case, the sum is 7, which is less than sqrt(61). So, does that mean there are no points z that satisfy this condition? Or maybe it's the other way around? Let me check the triangle inequality. The triangle inequality states that |z - 6i| + |z - 5| ≥ |(z - 5) - (z - 6i)| = | -5 + 6i | = sqrt(25 + 36) = sqrt(61). So, the sum of the distances must be at least sqrt(61). But the problem says the sum is 7, which is less than sqrt(61). That means there are no points z that satisfy this condition. So, the problem might have a typo or something.But the problem is given, so maybe I need to adjust my approach. Perhaps the sum is supposed to be sqrt(61), which would make sense because then it would be the minimal sum, and the set of points would be just the line segment between (0, 6) and (5, 0). In that case, to find the maximum |z|, I need to find the point on this line segment that is farthest from the origin.Alternatively, if the sum is indeed 7, which is less than sqrt(61), then there are no solutions, and the problem might be incorrect. But since the problem is asking for the maximum |z|, I think the intended condition is that the sum is sqrt(61), which would make the set of points the line segment between (0, 6) and (5, 0). So, I'll proceed with that assumption.Now, to find the maximum |z| on this line segment, I need to find the point on the line segment between (0, 6) and (5, 0) that is farthest from the origin. The farthest point would be one of the endpoints because the origin is not on the line segment, so the maximum distance should occur at one of the endpoints.Let me calculate |z| for both endpoints. For (0, 6), |z| is sqrt(0^2 + 6^2) = 6. For (5, 0), |z| is sqrt(5^2 + 0^2) = 5. So, the maximum |z| is 6.But wait, let me double-check. Maybe there's a point on the line segment that is farther from the origin than both endpoints. To check that, I can parametrize the line segment and find the maximum |z|.Let me parametrize the line segment from (0, 6) to (5, 0). Let t be a parameter between 0 and 1. Then, the coordinates can be written as (5t, 6 - 6t). So, z = 5t + (6 - 6t)i.Then, |z| = sqrt[(5t)^2 + (6 - 6t)^2] = sqrt[25t^2 + 36 - 72t + 36t^2] = sqrt[61t^2 - 72t + 36].To find the maximum of this function on t ∈ [0, 1], I can take the derivative and set it to zero.Let f(t) = 61t^2 - 72t + 36. Then, df/dt = 122t - 72. Setting this equal to zero gives t = 72/122 = 36/61 ≈ 0.590.Now, I need to check the value of |z| at t = 0, t = 1, and t = 36/61.At t = 0: |z| = sqrt[0 + 36] = 6.At t = 1: |z| = sqrt[61 - 72 + 36] = sqrt[25] = 5.At t = 36/61: Let's compute f(36/61):f(36/61) = 61*(36/61)^2 - 72*(36/61) + 36.First, compute (36/61)^2 = 1296/3721.Then, 61*(1296/3721) = 1296/61 ≈ 21.2459.Next, 72*(36/61) = 2592/61 ≈ 42.4918.So, f(36/61) ≈ 21.2459 - 42.4918 + 36 ≈ 14.7541.Then, |z| ≈ sqrt(14.7541) ≈ 3.84.Wait, that's less than both 5 and 6. So, the maximum |z| occurs at t = 0, which is 6.Therefore, the maximum value of |z| is 6.But wait, earlier I thought that if the sum is 7, which is less than sqrt(61), there are no solutions. But assuming the sum is sqrt(61), which makes sense because that's the minimal sum, then the maximum |z| is 6.I think the problem might have a typo, and the sum should be sqrt(61) instead of 7. Otherwise, there are no solutions. But since the problem is given, I'll proceed with the assumption that the sum is sqrt(61), and the maximum |z| is 6.Alternatively, if the sum is indeed 7, then there are no points z that satisfy the condition, so the maximum |z| would be undefined or non-existent. But since the problem is asking for the maximum, I think the intended condition is that the sum is sqrt(61), leading to the maximum |z| being 6.

💡Okay, so I have this problem where I need to find ( f(100) ) given that ( f(a + b) = f(a) + f(b) + ab ) and ( f(75) - f(51) = 1230 ). Hmm, functional equations can be tricky, but let me try to break it down step by step.First, the functional equation is ( f(a + b) = f(a) + f(b) + ab ). This reminds me of Cauchy's functional equation, which is ( f(a + b) = f(a) + f(b) ). But here, there's an extra term ( ab ). Maybe I can manipulate this equation to find a general form for ( f ).Let me try plugging in some specific values to see if I can find a pattern or figure out what ( f ) looks like. A common strategy is to set one of the variables to zero. Let me set ( b = 0 ):( f(a + 0) = f(a) + f(0) + a cdot 0 )Simplifying, that becomes:( f(a) = f(a) + f(0) )Subtracting ( f(a) ) from both sides, I get:( 0 = f(0) )So, ( f(0) = 0 ). That's a good start.Next, maybe I can set ( b = 1 ) to see how ( f ) behaves when we increment the input by 1. Let's try that:( f(a + 1) = f(a) + f(1) + a cdot 1 )Simplifying:( f(a + 1) = f(a) + f(1) + a )This is a recursive relation. It tells me that each time I increase the input by 1, the function increases by ( f(1) + a ). Maybe I can use this to express ( f ) in terms of ( f(1) ).Let me try to compute ( f(2) ) using this relation. If I set ( a = 1 ):( f(1 + 1) = f(1) + f(1) + 1 cdot 1 )So,( f(2) = 2f(1) + 1 )Similarly, ( f(3) ) would be:( f(3) = f(2 + 1) = f(2) + f(1) + 2 cdot 1 )Substituting ( f(2) ):( f(3) = (2f(1) + 1) + f(1) + 2 = 3f(1) + 3 )Hmm, I see a pattern here. Let me compute a few more terms to see if I can generalize.( f(4) = f(3 + 1) = f(3) + f(1) + 3 cdot 1 = (3f(1) + 3) + f(1) + 3 = 4f(1) + 6 )( f(5) = f(4 + 1) = f(4) + f(1) + 4 cdot 1 = (4f(1) + 6) + f(1) + 4 = 5f(1) + 10 )Wait a second, the coefficients of ( f(1) ) are just increasing by 1 each time, and the constants seem to be following a pattern as well. Let me list them out:- ( f(1) = f(1) )- ( f(2) = 2f(1) + 1 )- ( f(3) = 3f(1) + 3 )- ( f(4) = 4f(1) + 6 )- ( f(5) = 5f(1) + 10 )Looking at the constants: 0, 1, 3, 6, 10... These are triangular numbers! The nth triangular number is given by ( frac{n(n - 1)}{2} ). Let me check:- For ( f(2) ), the constant is 1, which is ( frac{2 cdot 1}{2} = 1 )- For ( f(3) ), the constant is 3, which is ( frac{3 cdot 2}{2} = 3 )- For ( f(4) ), the constant is 6, which is ( frac{4 cdot 3}{2} = 6 )- For ( f(5) ), the constant is 10, which is ( frac{5 cdot 4}{2} = 10 )Yes, that seems to fit. So, it looks like for any integer ( n ), ( f(n) = nf(1) + frac{n(n - 1)}{2} ). Let me test this formula with the values I have:- For ( n = 1 ): ( f(1) = 1f(1) + frac{1 cdot 0}{2} = f(1) ). Correct.- For ( n = 2 ): ( f(2) = 2f(1) + frac{2 cdot 1}{2} = 2f(1) + 1 ). Correct.- For ( n = 3 ): ( f(3) = 3f(1) + frac{3 cdot 2}{2} = 3f(1) + 3 ). Correct.Great, so this formula seems to hold. Now, the problem gives me ( f(75) - f(51) = 1230 ). Let me use my formula to express ( f(75) ) and ( f(51) ):( f(75) = 75f(1) + frac{75 cdot 74}{2} )( f(51) = 51f(1) + frac{51 cdot 50}{2} )Subtracting these:( f(75) - f(51) = (75f(1) + frac{75 cdot 74}{2}) - (51f(1) + frac{51 cdot 50}{2}) )Simplify the terms:First, the ( f(1) ) terms:( 75f(1) - 51f(1) = 24f(1) )Now, the constants:( frac{75 cdot 74}{2} - frac{51 cdot 50}{2} )Let me compute each part:( frac{75 cdot 74}{2} = frac{5550}{2} = 2775 )( frac{51 cdot 50}{2} = frac{2550}{2} = 1275 )Subtracting these:( 2775 - 1275 = 1500 )So, putting it all together:( f(75) - f(51) = 24f(1) + 1500 )But the problem states that this difference is 1230:( 24f(1) + 1500 = 1230 )Let me solve for ( f(1) ):Subtract 1500 from both sides:( 24f(1) = 1230 - 1500 )( 24f(1) = -270 )Divide both sides by 24:( f(1) = frac{-270}{24} )Simplify the fraction:Divide numerator and denominator by 6:( f(1) = frac{-45}{4} = -11.25 )Okay, so ( f(1) = -11.25 ). Now, I need to find ( f(100) ). Using the formula I derived earlier:( f(100) = 100f(1) + frac{100 cdot 99}{2} )Plugging in ( f(1) = -11.25 ):First, compute ( 100f(1) ):( 100 times (-11.25) = -1125 )Next, compute ( frac{100 cdot 99}{2} ):( frac{9900}{2} = 4950 )So, adding these together:( f(100) = -1125 + 4950 = 3825 )Therefore, ( f(100) = 3825 ).Wait, let me double-check my calculations to make sure I didn't make a mistake.First, ( f(75) - f(51) = 24f(1) + 1500 = 1230 ). So, ( 24f(1) = 1230 - 1500 = -270 ), which gives ( f(1) = -270 / 24 = -11.25 ). That seems correct.Then, ( f(100) = 100f(1) + (100 times 99)/2 = 100(-11.25) + 4950 = -1125 + 4950 = 3825 ). Yep, that checks out.So, I think I did everything correctly. The key was recognizing the pattern in the functional equation and expressing ( f(n) ) in terms of ( f(1) ). Once I had that, plugging in the given difference allowed me to solve for ( f(1) ), and then I could compute ( f(100) ).I wonder if there's another way to approach this problem, maybe by assuming a quadratic form for ( f ). Let me think about that for a moment.If I suppose that ( f(n) ) is a quadratic function, say ( f(n) = an^2 + bn + c ), then I can plug this into the functional equation and see if it works.So, ( f(a + b) = a(a + b)^2 + b(a + b) + c )Expanding:( f(a + b) = a(a^2 + 2ab + b^2) + b(a + b) + c = a^3 + 2a^2b + ab^2 + ab + b^2 + c )On the other hand, ( f(a) + f(b) + ab = [a a^2 + b a + c] + [a b^2 + b b + c] + ab )Wait, hold on, if ( f(n) = an^2 + bn + c ), then ( f(a) = a a^2 + b a + c )? No, that doesn't make sense because ( f(a) ) should be ( a a^2 + b a + c ), but ( a ) is a variable here, not a constant.Wait, maybe I confused the notation. Let me clarify. Let me denote the function as ( f(n) = kn^2 + mn + p ), where ( k ), ( m ), and ( p ) are constants to be determined.Then, ( f(a + b) = k(a + b)^2 + m(a + b) + p = k(a^2 + 2ab + b^2) + m(a + b) + p = k a^2 + 2k ab + k b^2 + m a + m b + p )On the other hand, ( f(a) + f(b) + ab = [k a^2 + m a + p] + [k b^2 + m b + p] + ab = k a^2 + k b^2 + m a + m b + 2p + ab )Now, set these equal:( k a^2 + 2k ab + k b^2 + m a + m b + p = k a^2 + k b^2 + m a + m b + 2p + ab )Subtracting the right side from the left side:( 2k ab + p - 2p = ab )Simplify:( 2k ab - p = ab )This must hold for all ( a ) and ( b ), so the coefficients of ( ab ) and the constants must match.So, for the ( ab ) term:( 2k = 1 ) => ( k = 1/2 )For the constant term:( -p = 0 ) => ( p = 0 )So, ( f(n) = frac{1}{2}n^2 + mn ). Wait, but earlier I had ( f(n) = nf(1) + frac{n(n - 1)}{2} ). Let me see if these are consistent.Given ( f(n) = frac{1}{2}n^2 + mn ), let's compute ( f(1) ):( f(1) = frac{1}{2}(1)^2 + m(1) = frac{1}{2} + m )From my earlier formula, ( f(n) = nf(1) + frac{n(n - 1)}{2} ). Let's compute ( f(1) ) using that:( f(1) = 1 cdot f(1) + frac{1 cdot 0}{2} = f(1) ). Hmm, that's just an identity, doesn't give new info.But if I express ( f(n) ) as ( frac{1}{2}n^2 + mn ), then:( f(n) = frac{1}{2}n^2 + mn = n(m + frac{n}{2}) )Comparing this to my earlier expression ( f(n) = nf(1) + frac{n(n - 1)}{2} ):( nf(1) + frac{n(n - 1)}{2} = n(m + frac{n}{2}) )Expanding both sides:Left side: ( nf(1) + frac{n^2 - n}{2} )Right side: ( mn + frac{n^2}{2} )Set them equal:( nf(1) + frac{n^2}{2} - frac{n}{2} = mn + frac{n^2}{2} )Subtract ( frac{n^2}{2} ) from both sides:( nf(1) - frac{n}{2} = mn )Factor out ( n ):( n(f(1) - frac{1}{2}) = mn )Since this must hold for all ( n ), we can divide both sides by ( n ) (assuming ( n neq 0 )):( f(1) - frac{1}{2} = m )So, ( m = f(1) - frac{1}{2} )Therefore, the quadratic form is consistent with my earlier expression. So, whether I approach it by recursion or by assuming a quadratic form, I end up with the same general solution.Given that, I can be more confident that my earlier result is correct. So, ( f(100) = 3825 ).I think I've covered all the steps and checked my work. I don't see any mistakes in my reasoning, so I feel confident that the answer is correct.**Final Answer**The value of ( f(100) ) is boxed{3825}.

💡First, I recognize that the dividend is 176, the divisor is 19, and the remainder is 5.To find the quotient, I subtract the remainder from the dividend: 176 minus 5 equals 171.Next, I divide this result by the divisor: 171 divided by 19 equals 9.Therefore, the quotient is 9.

💡Alright, so I've got this probability problem here, and I need to figure out the chance that the sum of white balls in Box A and red balls in Box B isn't a prime number. Let me try to break this down step by step.First, let's understand the setup. There's a bag with 6 red balls and 8 white balls, making a total of 14 balls. We're going to randomly put 5 balls into Box A and the remaining 9 into Box B. The question is about the probability that the sum of white balls in Box A and red balls in Box B is not a prime number.Okay, so let's denote the number of white balls in Box A as ( m ) and the number of red balls in Box B as ( n ). Since there are 8 white balls in total, the number of white balls in Box B would be ( 8 - m ). Similarly, since there are 6 red balls in total, the number of red balls in Box A would be ( 6 - n ).Now, the sum we're interested in is ( m + n ). We need to find the probability that this sum is not a prime number. To do this, it might be easier to first find the probability that ( m + n ) is a prime number and then subtract that from 1.So, let's figure out the possible values of ( m ) and ( n ). Since we're putting 5 balls into Box A, ( m ) can range from 0 to 5 (since there are only 8 white balls, but we can't have more than 5 in Box A). Similarly, ( n ) can range from 0 to 6 (since there are 6 red balls in total).But we also have some constraints based on the total number of balls. The number of balls in Box A is 5, so the number of red balls in Box A would be ( 5 - m ). Similarly, the number of white balls in Box B would be ( 8 - m ), and the number of red balls in Box B is ( n ). Since Box B has 9 balls, we have ( (8 - m) + n = 9 ). Simplifying this, we get ( n = m + 1 ).So, ( n ) is always one more than ( m ). That means the sum ( m + n ) is ( m + (m + 1) = 2m + 1 ). Therefore, the sum is always an odd number because it's twice an integer plus one. Now, let's list the possible values of ( m ) and the corresponding ( n ) and the sum ( 2m + 1 ):- If ( m = 0 ), then ( n = 1 ), sum = 1- If ( m = 1 ), then ( n = 2 ), sum = 3- If ( m = 2 ), then ( n = 3 ), sum = 5- If ( m = 3 ), then ( n = 4 ), sum = 7- If ( m = 4 ), then ( n = 5 ), sum = 9- If ( m = 5 ), then ( n = 6 ), sum = 11Now, let's identify which of these sums are prime numbers:- 1 is not a prime number.- 3 is a prime number.- 5 is a prime number.- 7 is a prime number.- 9 is not a prime number.- 11 is a prime number.So, the sums that are prime numbers occur when ( m = 1, 2, 3, 5 ). The sums that are not prime occur when ( m = 0, 4 ).Therefore, the probability that the sum is not prime is the probability that ( m = 0 ) or ( m = 4 ).Now, let's calculate the number of ways to choose the balls for each case.The total number of ways to choose 5 balls out of 14 is ( binom{14}{5} ).For ( m = 0 ): We choose 0 white balls and 5 red balls. The number of ways is ( binom{8}{0} times binom{6}{5} ).For ( m = 4 ): We choose 4 white balls and 1 red ball. The number of ways is ( binom{8}{4} times binom{6}{1} ).Let's compute these:- ( binom{8}{0} = 1 )- ( binom{6}{5} = 6 )- ( binom{8}{4} = 70 )- ( binom{6}{1} = 6 )So, the number of favorable outcomes for ( m = 0 ) is ( 1 times 6 = 6 ).The number of favorable outcomes for ( m = 4 ) is ( 70 times 6 = 420 ).Adding these together, the total number of favorable outcomes is ( 6 + 420 = 426 ).The total number of possible outcomes is ( binom{14}{5} = 2002 ).Therefore, the probability that the sum is not prime is ( frac{426}{2002} ).Simplifying this fraction, we can divide numerator and denominator by 2:( frac{213}{1001} )So, the probability is ( frac{213}{1001} ).

💡Alright, so I have this problem where there are 10 boys and 7 girls lining up randomly when the gym teacher whistles. I need to find the expected number of girls standing to the left of all the boys. Hmm, okay, let me try to wrap my head around this.First, let me visualize the situation. There are 17 students in total—10 boys and 7 girls. When they line up randomly, each permutation of these 17 students is equally likely. I need to figure out, on average, how many girls will be positioned to the left of every single boy in the line.So, to rephrase, I want to find the expected number of girls that are all the way to the left before any boys start appearing in the line. That is, if I look at the line from left to right, how many girls are there before I encounter the first boy. But wait, no, actually, the problem says "standing to the left of all the boys." So, it's not just the number of girls before the first boy, but all girls who are to the left of every boy in the entire line.Hmm, that's a bit different. So, it's not just the girls before the first boy, but any girl who is positioned to the left of all boys in the entire line. That would mean that all these girls are somewhere in the line, but each of them is to the left of every single boy. So, essentially, all these girls are clustered to the far left of the line, before any boys start.Wait, but in reality, the girls and boys are mixed randomly. So, it's possible that some girls are to the left of all boys, some are in the middle, and some are to the right. But the problem is specifically asking for the expected number of girls who are to the left of all boys. So, these are the girls who are positioned such that every boy is to their right.Okay, so how can I model this? Maybe I can think about each girl individually and calculate the probability that she is to the left of all boys. Then, since expectation is linear, I can sum these probabilities to get the expected number of such girls.Let me try that approach. For each girl, what is the probability that she is to the left of all boys? Well, considering all 17 students, each position is equally likely for any student. So, for a specific girl, the probability that she is to the left of all boys is the same as the probability that she is the first among all the boys and herself.Wait, no, that's not quite right. Because there are 10 boys, so for a specific girl to be to the left of all boys, she needs to be positioned before all 10 boys. So, in other words, among the 11 positions occupied by the 10 boys and this specific girl, she needs to be the first one.So, the probability that a specific girl is to the left of all boys is 1 divided by the number of boys plus one, which is 11. So, for each girl, the probability is 1/11.Therefore, since there are 7 girls, the expected number of girls to the left of all boys is 7 times 1/11, which is 7/11.Wait, that seems too straightforward. Let me verify this reasoning.Suppose I have n boys and m girls. For each girl, the probability that she is to the left of all boys is 1/(n+1). Therefore, the expected number of such girls is m/(n+1). In this case, n is 10 and m is 7, so 7/(10+1) = 7/11.That makes sense. But let me think about it another way to confirm.Imagine arranging all 17 students in a random permutation. The relative order of the 10 boys and 7 girls is equally likely to be any permutation. Now, consider the positions of the girls relative to the boys.Each girl has an equal chance of being in any position relative to the boys. So, for each girl, the probability that she is the first among all the boys and herself is 1/(10+1) = 1/11. Since there are 7 girls, the expected number is 7*(1/11) = 7/11.Alternatively, I can think of the problem as assigning ranks to each student. The student with the lowest rank is the first in line, and so on. For each girl, the probability that her rank is lower than all the boys' ranks is 1/11, as there are 11 students (10 boys + 1 girl) competing for the lowest rank.Therefore, summing over all 7 girls, the expected number is 7/11.Wait, but let me consider another perspective. Suppose I fix the positions of the boys and then see where the girls can be. If I fix the boys in certain positions, the girls can be in the remaining positions. But this might complicate things because the boys' positions are also random.Alternatively, I can think of the problem as a uniform random permutation of 17 elements, where 10 are labeled 'B' and 7 are labeled 'G'. I want to count the number of 'G's that appear before all 'B's. The expectation of this count is what I'm after.In such a permutation, the probability that a particular 'G' appears before all 'B's is equal to the probability that, among the 11 elements consisting of that 'G' and all 10 'B's, the 'G' is the first one. Since all permutations are equally likely, this probability is 1/11.Therefore, by linearity of expectation, the expected number of 'G's before all 'B's is 7*(1/11) = 7/11.Yes, this seems consistent across different ways of thinking about it. So, I think the expected number is indeed 7/11.Just to be thorough, let me consider a smaller example to see if this reasoning holds.Suppose there are 2 boys and 1 girl. What is the expected number of girls to the left of all boys?Using the same logic, the probability that the girl is to the left of both boys is 1/(2+1) = 1/3. Therefore, the expected number is 1*(1/3) = 1/3.Let's enumerate all possible permutations:1. G, B, B2. B, G, B3. B, B, GIn permutation 1, the girl is to the left of all boys, so count = 1.In permutations 2 and 3, the girl is not to the left of all boys, so count = 0.Therefore, the expected number is (1 + 0 + 0)/3 = 1/3, which matches our earlier result.Another example: 1 boy and 2 girls.Probability that a specific girl is to the left of the boy is 1/(1+1) = 1/2. Therefore, expected number is 2*(1/2) = 1.Enumerate permutations:1. G, G, B2. G, B, G3. B, G, GIn permutation 1, both girls are to the left of the boy, so count = 2.In permutations 2 and 3, only one girl is to the left of the boy, so count = 1.Wait, hold on, that doesn't seem to match. Let me recast it.Wait, actually, in permutation 1: G, G, B – both girls are to the left of the boy, so count = 2.In permutation 2: G, B, G – the first girl is to the left of the boy, the second girl is to the right, so count = 1.In permutation 3: B, G, G – the first position is a boy, so no girls are to the left of all boys, count = 0.Wait, that's different. So, the counts are 2, 1, 0.Therefore, the expected number is (2 + 1 + 0)/6 = 3/6 = 1/2.But according to our formula, it should be 2*(1/2) = 1. Hmm, discrepancy here.Wait, maybe my formula is incorrect? Or perhaps I made a mistake in interpreting the problem.Wait, in the original problem, we have 10 boys and 7 girls, and we want the number of girls to the left of all boys. In the smaller example with 1 boy and 2 girls, the expected number should be 1, but when I enumerate, I get 1/2.Hmm, that suggests that my initial reasoning might be flawed.Wait, let's think again. In the case of 1 boy and 2 girls, what exactly are we counting? The number of girls to the left of all boys. Since there's only 1 boy, it's the number of girls to the left of that boy.In permutation 1: G, G, B – both girls are to the left of the boy, so count = 2.In permutation 2: G, B, G – one girl is to the left of the boy, count = 1.In permutation 3: B, G, G – no girls are to the left of the boy, count = 0.So, the counts are 2, 1, 0, and since there are 3! = 6 permutations, but actually, the number of unique arrangements is 3: GGB, GBG, BGG. Wait, no, actually, for 1 boy and 2 girls, the number of distinct permutations is 3: GGB, GBG, BGG.Wait, but in reality, since the girls are indistinct? No, actually, the girls are distinct individuals, so the number of permutations is 3! / (2!1!) = 3, but considering the girls as distinct, it's 3! = 6.Wait, this is getting confusing. Maybe I should treat the girls as distinguishable.Let me label the girls as G1 and G2, and the boy as B.So, the permutations are:1. G1, G2, B2. G1, B, G23. G2, G1, B4. G2, B, G15. B, G1, G26. B, G2, G1Now, for each permutation, count the number of girls to the left of all boys.In permutations 1 and 3: Both girls are to the left of the boy, so count = 2.In permutations 2 and 4: One girl is to the left of the boy, count = 1.In permutations 5 and 6: No girls are to the left of the boy, count = 0.So, the counts are 2, 1, 2, 1, 0, 0.Therefore, the expected number is (2 + 1 + 2 + 1 + 0 + 0)/6 = 6/6 = 1.Ah, okay, so that matches our initial formula: number of girls times probability, which is 2*(1/2) = 1.So, in this case, the expectation is indeed 1, which matches both the formula and the enumeration.Earlier, I think I made a mistake by considering the girls as indistinct, which led to confusion. When treating them as distinct, the expectation comes out correctly.Therefore, going back to the original problem, with 10 boys and 7 girls, the expected number of girls to the left of all boys is 7*(1/11) = 7/11.Just to ensure, let me think of another small example. Suppose 3 boys and 2 girls.Using the formula, expected number is 2*(1/4) = 1/2.Let's enumerate:Label boys as B1, B2, B3 and girls as G1, G2.Total permutations: 5! = 120, but considering indistinctness, it's 5! / (3!2!) = 10.But treating them as distinct, it's 5! = 120.But for simplicity, let's treat them as distinct.Wait, actually, it's getting too complicated. Maybe I can think in terms of relative order.For each girl, the probability that she is to the left of all boys is 1/(number of boys + 1) = 1/4.Therefore, expected number is 2*(1/4) = 1/2.If I consider all possible arrangements, the number of girls to the left of all boys can be 0, 1, or 2.Calculating the exact expectation would require summing over all permutations, but given the linearity of expectation, it's sufficient to calculate the probability for each girl and sum them up.Therefore, I think the formula holds.So, in conclusion, for the original problem, the expected number of girls standing to the left of all boys is 7/11.