Answer:
[tex]Number\ of\ Committee = 6914880\ ways[/tex]
Step-by-step explanation:
Given
Total:
[tex]Freshmen= 7\\ Sophomores = 8\\ Juniors = 8\\ Seniors = 8[/tex]
Selection
[tex]Freshmen= 2\\ Sophomores = 3\\ Juniors = 4\\ Seniors = 5[/tex]
Required
Determine the number of selection
To do this, we make use of combination formula:
[tex]^nC_r = \frac{n!}{(n-r)!r!}[/tex]
For Freshmen, we have:
[tex]n = 7; r = 2[/tex]
[tex]^7C_2 = \frac{7!}{(7-2)!2!}[/tex]
[tex]^7C_2 = \frac{7!}{5!2!}[/tex]
[tex]^7C_2 = \frac{7 * 6 * 5!}{5! * 2 * 1}[/tex]
[tex]^7C_2 = \frac{7 * 6}{2}[/tex]
[tex]^7C_2 = \frac{42}{2}[/tex]
[tex]^7C_2 = 21[/tex]
For Sophomores, we have:
[tex]n =9;r=3[/tex]
[tex]^9C_3 = \frac{9!}{(9-3)!3!}[/tex]
[tex]^9C_3 = \frac{9!}{6!3!}[/tex]
[tex]^9C_3 = \frac{9 * 8 * 7 * 6!}{6! * 3 * 2 * 1}[/tex]
[tex]^9C_3 = \frac{9 * 8 * 7 }{6}[/tex]
[tex]^9C_3 = \frac{504}{6}[/tex]
[tex]^9C_3 = 84[/tex]
For Juniors, we have:
[tex]n = 8; r = 4[/tex]
[tex]^8C_4 = \frac{8!}{(8-4)!4!}[/tex]
[tex]^8C_4 = \frac{8!}{4!4!}[/tex]
[tex]^8C_4 = \frac{8 * 7 *6 * 5 * 4!}{4!*4 * 3 * 2 * 1}[/tex]
[tex]^8C_4 = \frac{8 * 7 *6 * 5}{4 * 3 * 2 * 1}[/tex]
[tex]^8C_4 = \frac{1680}{24}[/tex]
[tex]^8C_4 = 70[/tex]
For Seniors
[tex]n = 8; r = 5[/tex]
[tex]^8C_5 = \frac{8!}{(8-5)!5!}[/tex]
[tex]^8C_5 = \frac{8!}{3!5!}[/tex]
[tex]^8C_5 = \frac{8 * 7 *6 * 5!}{5! * 3 * 2 * 1}[/tex]
[tex]^8C_5 = \frac{8 * 7 *6}{3 * 2 * 1}[/tex]
[tex]^8C_5 = \frac{8 * 7}{1}[/tex]
[tex]^8C_5 = 56[/tex]
[tex]Number\ of\ Committee = Freshmen\ and\ Sophomores\ and\ Juniors\ and\ Seniors[/tex]
and implies *, so we have:
[tex]Number\ of\ Committee = 21 * 84 * 70 * 56[/tex]
[tex]Number\ of\ Committee = 6914880\ ways[/tex]
