A student representative council consists of 8 girls and 6 boys. If an editorial board consisting of 5 persons is to be formed, what is the probability that the board consists of
(a) 3 girls and 2 boys ;
(b) either all girls or all boys.
(a) 3 girls and 2 boys ;
(b) either all girls or all boys.
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Correct Answer: Option n
Explanation:
Total number of board members = 14 members.
Number of editorial board members arrangement possible without restriction = \(^{14}C_{5}\) ways
= \(\frac{14!}{5! (14 - 5)!} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2}\)
= \(\text{2002 ways}\).
(a) With 3 girls and 2 boys
= \(^{8}C_{3} \times ^{6}C_{2} \implies \frac{8!}{3! (8 - 3)!} \times \frac{6!}{2! (6 - 2)!}\)
= \(56 \times 15 = 840\)
p(3 girls and 2 boys) = \(\frac{840}{2002}\)
= \(\frac{60}{143}\)
(b) Editorial board with all girls = \(^{8}C_{5}\) ways
= \(\frac{8!}{5! (8 - 5)!} = 56\)
p(all girls ) = \(\frac{56}{2002}\)
With all boys = \(^{6}C_{5}\) ways
= \(\frac{6!}{5! (6 - 5)!} = 6\)
p(all boys) = \(\frac{6}{2002}\)
p(either all boys or all girls) = \(\frac{56}{2002} + \frac{6}{2002} = \frac{31}{1001}\).
Total number of board members = 14 members.
Number of editorial board members arrangement possible without restriction = \(^{14}C_{5}\) ways
= \(\frac{14!}{5! (14 - 5)!} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2}\)
= \(\text{2002 ways}\).
(a) With 3 girls and 2 boys
= \(^{8}C_{3} \times ^{6}C_{2} \implies \frac{8!}{3! (8 - 3)!} \times \frac{6!}{2! (6 - 2)!}\)
= \(56 \times 15 = 840\)
p(3 girls and 2 boys) = \(\frac{840}{2002}\)
= \(\frac{60}{143}\)
(b) Editorial board with all girls = \(^{8}C_{5}\) ways
= \(\frac{8!}{5! (8 - 5)!} = 56\)
p(all girls ) = \(\frac{56}{2002}\)
With all boys = \(^{6}C_{5}\) ways
= \(\frac{6!}{5! (6 - 5)!} = 6\)
p(all boys) = \(\frac{6}{2002}\)
p(either all boys or all girls) = \(\frac{56}{2002} + \frac{6}{2002} = \frac{31}{1001}\).