Question

The senior class at a very small high school has 25 students. Officers need to be elected for four positions: President, Vice-President, Secretary, and Treasurer. a. In how many ways can the four officers be chosen? b. If there are 13 girls and 12 boys in the class, in how many ways can the officers be chosen if the President and Treasurer are girls and the Vice-President and Secretary are boys?

Answer 1

(a) The total number of ways to select 4 officers from from 25 students is 12,650.

(b) The total number of ways the four officers are selected such that the President and Treasurer are girls and the Vice-President and Secretary are boys is 5,148.

Explanation:

(a)

It is provided that there are a total of n = 25 students.

Officers need to be elected for four positions:

President, Vice-President, Secretary, and Treasurer.

k = 4

In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.

The formula to compute the combinations of k items from n is given by the formula:

`{n\choose k}=(n!)/(k!(n-k)!)`

Compute the number of ways to select 4 students from from 25 as follows:

`{25\choose 4}=(25!)/(4!(25-4)!)`

`=(25!)/(4!* 21!)\\\\=(25* 24* 23* 22* 21!)/(4!* 21!)\\\\=(25* 24* 23* 22)/(4* 3* 2* 1)\\\\=12650`

Thus, the total number of ways to select 4 officers from from 25 students is 12,650.

(b)

It is provided that of the 25 students, there are 13 girls and 12 boys in the class.

For the post of President and Treasurer only girls are selected.

For the post of Vice-President and Secretary only boys are selected.

Compute the number of ways to select 2 girls for the post of President and Treasurer as follows:

`{13\choose 2}=(13!)/(2!(13-2)!)`

`=(13!)/(2!* 11!)\\\\=(13* 12* 11!)/(2!* 11!)\\\\=(13* 12)/( 2* 1)\\\\=78`

Compute the number of ways to select 2 boys for the post of Vice-President and Secretary as follows:

`{12\choose 2}=(12!)/(2!(12-2)!)`

`=(12!)/(2!* 10!)\\\\=(12* 11* 10!)/(2!* 10!)\\\\=(12* 11)/( 2* 1)\\\\=66`

The number of ways the four officers are selected such that the President and Treasurer are girls and the Vice-President and Secretary are boys is:

`{13\choose 2}* {12\choose 2}=78* 66=5148`

Thus, the total number of ways the four officers are selected such that the President and Treasurer are girls and the Vice-President and Secretary are boys is 5,148.