Question

Out of 6 boys and 4 girls, a committee of 5 members is to be formed. In how many ways can this be done, if at least 2 girls are included?

• A. 126
• B. 186
• C. 140
• D. 156

Hint:

When forming committees with restrictions (like having at least 2 girls), break the problem into cases based on the number of girls and boys, then add the results together.

Answer 1

The Correct Option is B

• Number of boys = 6
• Number of girls = 4
• Committee size = 5
• Condition: At least 2 girls must be included

Approach

We consider all possible cases that satisfy the condition of having at least 2 girls in the committee:

• 2 girls and 3 boys
• 3 girls and 2 boys
• 4 girls and 1 boy

We calculate each case separately and then sum the results.

Case 1: 2 Girls and 3 Boys

\[ \text{Number of ways to choose 2 girls} = \binom{4}{2} = 6 \]

\[ \text{Number of ways to choose 3 boys} = \binom{6}{3} = 20 \]

\[ \text{Total for this case} = 6 \times 20 = 120 \]

Case 2: 3 Girls and 2 Boys

\[ \text{Number of ways to choose 3 girls} = \binom{4}{3} = 4 \]

\[ \text{Number of ways to choose 2 boys} = \binom{6}{2} = 15 \]

\[ \text{Total for this case} = 4 \times 15 = 60 \]

Case 3: 4 Girls and 1 Boy

\[ \text{Number of ways to choose 4 girls} = \binom{4}{4} = 1 \]

\[ \text{Number of ways to choose 1 boy} = \binom{6}{1} = 6 \]

\[ \text{Total for this case} = 1 \times 6 = 6 \]

Total Number of Ways

\[ \text{Total} = 120 (\text{Case 1}) + 60 (\text{Case 2}) + 6 (\text{Case 3}) = 186 \]

Final Answer

The total number of ways to form the committee is 186.

Verification

We can verify by calculating the total possible committees without restrictions and subtracting the invalid cases:

\[ \text{Total possible committees} = \binom{10}{5} = 252 \]

\[ \text{Committees with 0 girls} = \binom{6}{5} = 6 \]

\[ \text{Committees with 1 girl} = \binom{4}{1} \times \binom{6}{4} = 4 \times 15 = 60 \]

\[ \text{Valid committees} = 252 - 6 - 60 = 186 \]

This confirms our previous calculation.