A committee of 4 members is to be formed from 7 men and 5 women. In how many ways can it be formed if the committee has at most one woman?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 210
"At most one woman" means the committee has no woman or exactly one woman. Each case is counted (choose the men and the women, multiply), and the cases are added.
No woman, 4 men: \(\displaystyle ^7C_4=\frac{7\times6\times5\times4}{4\times3\times2\times1}=35\)
1 woman and 3 men: \(^5C_1\times{}^7C_3=5\times35=175\)
Total \(=35+175=210\)
Why the other options are incorrect
Option (1) 175 — Incorrect. This counts only the committees with exactly one woman and leaves out the all-men committees.
Option (2) 495 — Incorrect. This is \(^{12}C_4\), every committee of 4, including those with 2, 3 or 4 women.
Option (4) 35 — Incorrect. This counts only the committees with no woman.