Question

A man has 9 friends: 4 boys and 5 girls. In how many ways can he invite them, if there have to be exactly 3 girls in the invitees?

• A. 320
• B. 160
• C. 80
• D. 200

Hint:

Break selection into independent parts — first choose required girls, then choose boys freely. Multiply the counts.

Answer 1

The Correct Option is B

We need to select exactly 3 girls out of 5 and the remaining (any number) from boys. But since the total number of invitees is not fixed, and only 3 girls must be included, the number of boys can be: 0, 1, 2, 3, or 4. Let’s calculate all valid combinations: \[ \text{Number of ways to choose 3 girls from 5} = \binom{5}{3} = 10 \] Now for each of the 5 possible numbers of boys (0 to 4): \[ \text{Total ways} = \sum_{r=0}^{4} \binom{4}{r} = \binom{4}{0} + \binom{4}{1} + \binom{4}{2} + \binom{4}{3} + \binom{4}{4} = 1 + 4 + 6 + 4 + 1 = 16 \] So total combinations: \[ 10 \times 16 = \boxed{160} \] \fbox{Final Answer: (B) 160}