Question

From a group of 6 girls and 7 boys, how many 5-member committees consist of: members of the same gender?

Answer 1

To form 5-member committees of the same gender from 6 girls and 7 boys, there are 6 possible committees of girls and 21 possible committees of boys, resulting in a total of 27 such committees.

Step-by-step explanation:

The question involves calculating the number of ways to form a 5-member committee from a group of 6 girls and 7 boys such that all committee members are of the same gender. This is a problem that involves combinations because the order in which we select the committee members does not matter. We will tackle this problem by treating the formation of committees with girls and boys as separate cases.

Committees of Girls

To calculate the number of possible 5-member committees consisting entirely of girls, we use the combination formula C(n, k) = n! / (k! * (n - k)!), where n is the total number we can choose from and k is the number we need to choose:

C(6, 5) = 6! / (5! * (6 - 5)!) = 6

Committees of Boys

To calculate the number of possible 5-member committees consisting entirely of boys, we use the same combination formula:

C(7, 5) = 7! / (5! * (7 - 5)!) = 21

To find the total number of 5-member committees that consist of members of the same gender, we simply add the two results together:

Total committees = Committees of girls + Committees of boys = 6 + 21 = 27

Answer 2

27 5-member committees consist of members of the same gender

Step-by-step explanation:

The order in which the members are chosen to the group is not important, so we use the combinations formula to solve this question.

Combinations formula:

`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

All girls:

5 girls from a set of 6. So

`C_(6,5) = (6!)/(5!(6-5)!) = 6`

All boys:

5 boys from a set of 7. So

`C_(7,5) = (7!)/(5!(7-5)!) = 21`

Total:

Either all girls or all boys.

6 + 21 = 27

27 5-member committees consist of members of the same gender