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

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is A

Let's analyze each statement to determine their validity:

Statement I: A committee of 4 can be made out of 5 men and 3 women containing at least one woman in 65 ways. 

To solve this, we use the concept of combinations. A committee of 4 members can be formed in different ways considering at least one woman must be present.

Total number of ways to choose 4 people from 8 (5 men + 3 women):

\(^8C_4 = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\)

Now, calculate the number of committees with no women (i.e., all men):

\(^5C_4 = \frac{5!}{4!(5-4)!} = \frac{5}{1} = 5\)

Therefore, committees with at least one woman:

\(^8C_4 - ^5C_4 = 70 - 5 = 65\)

Thus, Statement I is true.

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.

Identify the vowels and positions:

• Vowels in "ARRANGE" are A, A, E.
• Even positions available: 2nd, 4th, and 6th.

Place vowels in even positions. Since there are 3 vowels and 3 positions:

The permutations of A, A, E in these positions:

\(\frac{3!}{2!} = 3\) (since A repeats twice)

Now place consonants (R, R, N, G) in the remaining positions (1st, 3rd, 5th, 7th):

The permutations of R, R, N, G:

\(\frac{4!}{2!} = 12\) (since R repeats twice)

Total combinations by multiplying both possibilities together:

\(3 \times 12 = 36\)

Thus, Statement II is correct.

Therefore, both statements are true.