At the first tri-city meeting, there were 8 people from town \( A, 7 \) people from town B, and 5 people from town \( C \). If the council consists of 5 people, find the probability of 3 from town A and 2 from town B.

To find the probability of 3 people from town A and 2 people from town B in the council of 5 people, we need to calculate the total number of ways to choose 3 people from town A and 2 people from town B, and then divide it by the total number of ways to choose 5 people from the total of 20 people. Let's denote: - \( A \) as the number of people from town A, - \( B \) as the number of people from town B, - \( C \) as the number of people from town C, - \( T \) as the total number of people. Given: - \( A = 8 \), - \( B = 7 \), - \( C = 5 \), - \( T = A + B + C = 8 + 7 + 5 = 20 \). We need to find the probability of 3 people from town A and 2 people from town B in the council of 5 people. The total number of ways to choose 3 people from town A is \( \binom{8}{3} \) and the total number of ways to choose 2 people from town B is \( \binom{7}{2} \). The total number of ways to choose 5 people from the total of 20 people is \( \binom{20}{5} \). Therefore, the probability of 3 people from town A and 2 people from town B in the council of 5 people is: \[ P = \frac{\binom{8}{3} \times \binom{7}{2}}{\binom{20}{5}} \] Now, let's calculate the values of the binomial coefficients and then find the probability. Calculate the value by following steps: - step0: Calculate: \( { }_{8}C_{3}\) - step1: Expand the expression: \(\frac{8!}{3!\times \left(8-3\right)!}\) - step2: Subtract the numbers: \(\frac{8!}{3!\times 5!}\) - step3: Expand the expression: \(\frac{8\times 7\times 6\times 5\times 4\times 3!}{3!\times 5!}\) - step4: Reduce the fraction: \(\frac{8\times 7\times 6\times 5\times 4}{5!}\) - step5: Multiply the terms: \(\frac{6720}{5!}\) - step6: Calculate the value: \(\frac{6720}{120}\) - step7: Reduce the fraction: \(56\) Calculate or simplify the expression \( \binom{7}{2} \). Calculate the value by following steps: - step0: Calculate: \( { }_{7}C_{2}\) - step1: Expand the expression: \(\frac{7!}{2!\times \left(7-2\right)!}\) - step2: Subtract the numbers: \(\frac{7!}{2!\times 5!}\) - step3: Expand the expression: \(\frac{7\times 6\times 5\times 4\times 3\times 2!}{2!\times 5!}\) - step4: Reduce the fraction: \(\frac{7\times 6\times 5\times 4\times 3}{5!}\) - step5: Multiply the terms: \(\frac{2520}{5!}\) - step6: Calculate the value: \(\frac{2520}{120}\) - step7: Reduce the fraction: \(21\) Calculate or simplify the expression \( \binom{20}{5} \). Calculate the value by following steps: - step0: Calculate: \( { }_{20}C_{5}\) - step1: Expand the expression: \(\frac{20!}{5!\times \left(20-5\right)!}\) - step2: Subtract the numbers: \(\frac{20!}{5!\times 15!}\) - step3: Expand the expression: \(\frac{20\times 19\times 18\times 17\times 16\times 15!}{5!\times 15!}\) - step4: Reduce the fraction: \(\frac{20\times 19\times 18\times 17\times 16}{5!}\) - step5: Multiply the terms: \(\frac{1860480}{5!}\) - step6: Calculate the value: \(\frac{1860480}{120}\) - step7: Reduce the fraction: \(15504\) Calculate or simplify the expression \( 56*21/15504 \). Calculate the value by following steps: - step0: Calculate: \(\frac{56\times 21}{15504}\) - step1: Multiply the numbers: \(\frac{1176}{15504}\) - step2: Reduce the fraction: \(\frac{49}{646}\) The probability of 3 people from town A and 2 people from town B in the council of 5 people is \( \frac{49}{646} \).