APPLICATIONS

7 5 - 9 0 ■ Counting Principles Solve these exercises by using the appropriate counting principle(s).

Choosing a Group Sixteen boys and nine girls go on a camping trip. In how many ways can a group of six be selected to gather firewood, given the following conditions?

(a) The group consists of two girls and four boys.

(b) The group contains at at least two girls.

(a)

To find:

The number of ways a group of six to be selected to gather firewood.

Given:

There are 16 boys and 9 girls on the trip. A group of 6 is to be selected given that the group consists of two girls and four boys.

Approach:

The order for choosing 2 girls from a total of 9 girls and the order of choosing 4 boys from a total of 16 boys is not important.

Write the formula of number of combinations.

C(n,r)=n!r!(n−r)! …(1)

Here, n is the total number of objects and r is the number of objects chosen.

Calculation:

First substitute 9 for n and 2 for r in (1)

C(9,2)=9!2!(9−2)!=9×8×7!

(b)

To find:

The number of ways a group of six be selected to gather firewood.