Twenty-two candidates have filed for the upcoming county council election. 9 are

Question

Twenty-two candidates have filed for the upcoming county council election. 9 are women and 13 are men.

(a) In how many ways can 8 county council members be randomly elected out of the 22 candidates? Show work.

(b) In how many ways can 8 county council members be randomly elected from 22 candidates if 4 must be women and 4 must be men? Show work.

(c) If 8 county council members are randomly elected from 22 candidates, what is the probability that 4 are women and 4 are men? Round answer to nearest ten-thousandth (4 places after decimal). Show work.

Explanation / Answer

We are doing selection so we can use combination formula C(n,r)=n!/[r!*(n-r)!] where r people are selected from n people
Answer(a):
we need to select 8 members from 22 so answer will be
C(22,8)=22!/(8!*(22-8)!)=22!/(8!*14!)=319770
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Answer(b):
4 men can be selected out of 13 men in C(13,4) and
4 women can be selected out of 9 women in C(9,4)
So answer will be=
C(13,4)*C(9,4)=13!/[4!*(13-4)!]*9!/[4!*(9-4)!]=13!/[4!*9!]*13!/[4!*5!]=90090
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Answer(c):
From previous work we found that 8 members men or women can be selected in 319770 ways.
if 4 are men and 4 women then that can be done in 90090 ways.
So the required probability will be = 90090/319770=91/323 =0.2817

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