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

Answer:

$Number\ of\ Committee = 6914880\ ways$

Step-by-step 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:

$^nC_r = \frac{n!}{(n-r)!r!}$

For Freshmen, we have:

$n = 7; r = 2$

$^7C_2 = \frac{7!}{(7-2)!2!}$

$^7C_2 = \frac{7!}{5!2!}$

$^7C_2 = \frac{7 * 6 * 5!}{5! * 2 * 1}$

$^7C_2 = \frac{7 * 6}{2}$

$^7C_2 = \frac{42}{2}$

$^7C_2 = 21$

For Sophomores, we have:

$n =9;r=3$

$^9C_3 = \frac{9!}{(9-3)!3!}$

$^9C_3 = \frac{9!}{6!3!}$

$^9C_3 = \frac{9 * 8 * 7 * 6!}{6! * 3 * 2 * 1}$

$^9C_3 = \frac{9 * 8 * 7 }{6}$

$^9C_3 = \frac{504}{6}$

$^9C_3 = 84$

For Juniors, we have:

$n = 8; r = 4$

$^8C_4 = \frac{8!}{(8-4)!4!}$

$^8C_4 = \frac{8!}{4!4!}$

$^8C_4 = \frac{8 * 7 *6 * 5 * 4!}{4!*4 * 3 * 2 * 1}$

$^8C_4 = \frac{8 * 7 *6 * 5}{4 * 3 * 2 * 1}$

$^8C_4 = \frac{1680}{24}$

$^8C_4 = 70$

For Seniors

$n = 8; r = 5$

$^8C_5 = \frac{8!}{(8-5)!5!}$

$^8C_5 = \frac{8!}{3!5!}$

$^8C_5 = \frac{8 * 7 *6 * 5!}{5! * 3 * 2 * 1}$

$^8C_5 = \frac{8 * 7 *6}{3 * 2 * 1}$

$^8C_5 = \frac{8 * 7}{1}$

$^8C_5 = 56$

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

and implies *, so we have:

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

$Number\ of\ Committee = 6914880\ ways$