Question

A committee of five members is to be formed from among six boys and five girls. Find the number of ways of selecting the committee, if it is to consist of at least one boy and at least one girl?

• A. 455
• B. 456
• C. 461
• D. 477
• E. None of these

Answer 1

The Correct Option is A

Step 1: Understand the problem.
A committee of five members is to be formed from among six boys and five girls. We need to find the number of ways of selecting the committee, given that it must consist of at least one boy and at least one girl.

Step 2: Calculate the total number of ways to form the committee without any restrictions.
The total number of ways to select 5 members from a group of 11 people (6 boys + 5 girls) is given by the combination formula:
\( \binom{11}{5} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462 \)

Step 3: Subtract the cases where there are no boys or no girls in the committee.
- The number of ways to form a committee with no boys (only girls) is:
\( \binom{5}{5} = 1 \) (since all 5 members must be girls).
- The number of ways to form a committee with no girls (only boys) is:
\( \binom{6}{5} = 6 \) (since all 5 members must be boys).

Step 4: Calculate the number of valid committees with at least one boy and at least one girl.
The number of valid committees is the total number of committees minus the committees with only boys or only girls:
\( 462 - 1 - 6 = 455 \)

Step 5: Conclusion.
The number of ways of selecting the committee, with at least one boy and at least one girl, is 455.

Final Answer:
The correct option is (A): 455.