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Complete Question
Among the seven nominees for two vacancies on the city council are three men and four women. In how many ways may these vacancies be filled
a) with any two of the nominees?
b) with any two of the women?
c) with one of the men and one of the women?
Answer:
a) 21 ways
b) 6 ways
c) 12 ways
Step-by-step explanation:
We solve this question using combination formula
C(n, r) = nCr = n!/r! (n - r)!
a) with any two of the nominees?
Probability (two of the nominees) = 7C2
= 7!/2! ×(7 - 2)!
= 7!/ 2! × 5!
= 7 × 6 × 5 × 4 × 3 × 2 × 1/2 × 1 ×(5 × 4 × 3 × 2 × 1)
= 21
b) with any two of the women?
We have a total of 4 women
Hence, the probability of any two of the four women, filling the vacancies =
P(any two of the women) = 4C2
= 4!/2! ×( 4 - 2)!
= 4!/ 2! × 2!
= 4 × 3 × 2 × 1/ 2 × 1 ×( 2 × 1)
= 6
c) with one of the men and one of the
Total number of men = 3
Total number of women = 4
= 3C1 × 4C1
= [3!/1! ×(3 - 1)! ] × [4!/1! ×(4 - 1)! ]
= [3!/1! × 2!] × [4!/1! ×3!]
= [3 × 2 × 1/ 1 × 2 × 1] × [4 × 3 × 2 × 1/ 1 × 3 × 2 × 1]
= 3 × [24/6]
= 3 × 4
= 12 ways
