Question

Given below are two statements

Statement I : A committee of 4 can be made out of 5 men and 3 women containing at least one woman in 65 ways. Statement II : The number of words which can be formed using letters of the word 'ARRANGE' so that vowels always occupy even place is 36.

In light of the above statements, choose the correct answer from the options given below

Accepted Answer

A. Statement I: Total ways to form committee of 4 from 8 people = $\binom{8}{4} = 70$. Ways with no women (only men) = $\binom{5}{4} = 5$. Ways with at least 1 woman = $70 - 5 = 65$. True. Statement II: 'ARRANGE' has 7 positions (1, 2, 3, 4, 5, 6, 7), with 3 even positions (2, 4, 6). Vowels are A, A, E (3 vowels with two A's). Consonants are R, R, N, G (4 consonants with two R's). - Ways to place vowels in even positions = $\frac{3!}{2!} = 3$. - Ways to place consonants in odd positions = $\frac{4!}{2!} = 12$. Total words = $3 	imes 12 = 36$. True. Both statements are true.

Committee selection and word permutation statements

Explanation

Practise this under exam conditions