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iren2701 [21]
3 years ago
10

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.
Mathematics
1 answer:
m_a_m_a [10]3 years ago
6 0

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

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