Question

The College of Science Council has one student representative from each of the five science departments (biology.chemistry, geology. mathematics, physics). In how many ways can3. Both a council president and a vice president be selected?b. A president, a vice president, and a secretary be selected?c. Two members be selected for the Dean's Council?

Answer 1

Answer: a. 20 b . 60 . c. 10

Explanation:

• When the order of selection is important , we use Permutations.

The number of permutation of selecting r things from n things =
`^nP_r=(n!)/((n-r)!)`

• When the order of selection is not important , we use Combinations.

The number of permutation of selecting r things from n things =
`^nC_r=(n!)/(r!(n-r)!)`

Given : Number of science departments : n= 5

a . If both a council president and a vice president be selected , so we need to select for 2 positions (order matters).

Number of ways :
`^5P_2=(5!)/((5-2)!)`

`=(5*4*3!)/(3!)=20`

∴ The number of ways a council can select a president and a vice president=20 .

b. If a resident, a vice president, and a secretary be selected, then we need to select for 3 positions (order matters).

Number of ways :
`^5P_3=(5!)/((5-3)!)`

`=(5*4*3*2!)/(2!)=60`

∴ The number of ways a council can select a resident, a vice president, and a secretary =60.

c. If two members be selected for the Dean's Council , then order not mattes.

Number of ways :
`^5C_2=(5!)/(2!(5-2)!)`

`=(5*4*3!)/(2*3!)=10`

∴ The number of ways a council can select two members for the Dean's Council =10