Question

7 freshmen, 9 sophomores, 8 juniors, and 8 seniors are eligible to be on a committee. In how many ways can a dance committee of 14 students be chosen? In how many ways can a dance committee be chosen if it is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors.

Answer 1

`Number\ of\ Committee = 6914880\ ways`

Explanation:

Given

Total:

`Freshmen= 7\\ Sophomores = 8\\ Juniors = 8\\ Seniors = 8`

Selection

`Freshmen= 2\\ Sophomores = 3\\ Juniors = 4\\ Seniors = 5`

Required

Determine the number of selection

To do this, we make use of combination formula:

`^nC_r = (n!)/((n-r)!r!)`

For Freshmen, we have:

`n = 7; r = 2`

`^7C_2 = (7!)/((7-2)!2!)`

`^7C_2 = (7!)/(5!2!)`

`^7C_2 = (7 * 6 * 5!)/(5! * 2 * 1)`

`^7C_2 = (7 * 6)/(2)`

`^7C_2 = (42)/(2)`

`^7C_2 = 21`

For Sophomores, we have:

`n =9;r=3`

`^9C_3 = (9!)/((9-3)!3!)`

`^9C_3 = (9!)/(6!3!)`

`^9C_3 = (9 * 8 * 7 * 6!)/(6! * 3 * 2 * 1)`

`^9C_3 = (9 * 8 * 7 )/(6)`

`^9C_3 = (504)/(6)`

`^9C_3 = 84`

For Juniors, we have:

`n = 8; r = 4`

`^8C_4 = (8!)/((8-4)!4!)`

`^8C_4 = (8!)/(4!4!)`

`^8C_4 = (8 * 7 *6 * 5 * 4!)/(4!*4 * 3 * 2 * 1)`

`^8C_4 = (8 * 7 *6 * 5)/(4 * 3 * 2 * 1)`

`^8C_4 = (1680)/(24)`

`^8C_4 = 70`

For Seniors

`n = 8; r = 5`

`^8C_5 = (8!)/((8-5)!5!)`

`^8C_5 = (8!)/(3!5!)`

`^8C_5 = (8 * 7 *6 * 5!)/(5! * 3 * 2 * 1)`

`^8C_5 = (8 * 7 *6)/(3 * 2 * 1)`

`^8C_5 = (8 * 7)/(1)`

`^8C_5 = 56`

`Number\ of\ Committee = Freshmen\ and\ Sophomores\ and\ Juniors\ and\ Seniors`

and implies *, so we have:

`Number\ of\ Committee = 21 * 84 * 70 * 56`

`Number\ of\ Committee = 6914880\ ways`