Question

A teacher has 68 girls and 79 boys in all of her classes. A committee of 10 students is needed to represent the statistics of all classes. If exactly 5 girls and 5 boys are chosen at random, how many possible committees are possible?

a) 65,565
b) 7,980
c) 10,745
d) 9,561

Answer 1

To calculate the number of possible committees with exactly 5 girls and 5 boys chosen randomly from 68 girls and 79 boys, we use combinations: C(68, 5) for the girls and C(79, 5) for the boys, then multiply the results to get the total number of possible committees.

Step-by-step explanation:

To determine the number of possible committees formed from 68 girls and 79 boys where the committee is made up of exactly 5 girls and 5 boys, we can use the concept of combinations. To find the combination of 5 girls from 68, we calculate C(68, 5), and for 5 boys from 79, we calculate C(79, 5). The total number of possible committees is the product of these two combinations.

To calculate a combination, we use the formula C(n, k) = n! / (k! * (n-k)!), where n! is the factorial of n, and k is the number of items to choose.

Therefore, the number of ways to choose 5 girls from 68 is C(68, 5) and the number of ways to choose 5 boys from 79 is C(79, 5). The final answer is the product of these two values.

The calculations will give us the total number of possible distinct committees that can be formed under the given conditions.