What is the probability that the committee includes exactly 3 gentlemen?

Calculation:

Given,

A committee of 6 members is to be formed from a group of 7 gentlemen and 4 ladies.

The total number of ways to form the committee of 6 members is:

\( \binom{11}{6} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 462 \)

The number of ways to form a committee with exactly 3 gentlemen and 3 ladies is:

\( \binom{7}{3} \times \binom{4}{3} = 35 \times 4 = 140 \)

The probability of forming a committee with exactly 3 gentlemen is:

\( \text{Probability} = \frac{140}{462} = \frac{10}{33} \)

Hence, the correct answer is Option 1.