A panel of 4 is to be chosen from 8 men and 3 women. In how many ways can the panel be chosen if it has at most one woman?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 238
"At most one" means no woman or exactly one woman. The two cases cannot happen together, so their counts are added.
No woman: 4 men from 8, \(\displaystyle ^8C_4=\frac{8\times7\times6\times5}{4\times3\times2\times1}=70\)
One woman: \(^3C_1\times\,^8C_3=3\times56=168\)
Ways \(=70+168=238\)
Why the other options are incorrect
Option (1) 168 — Incorrect. This is only the case of exactly one woman; the case with no woman is left out.
Option (2) 330 — Incorrect. This is \(^{11}C_4\), every panel, including those with 2 or 3 women.
Option (4) 70 — Incorrect. This is only the case with no woman.