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.
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}).