A team of 5 is to be made of 3 men chosen from 7 men and 2 women chosen from 4 women. In how many ways can the team be made?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 210
The men and the women are chosen separately; every choice of men can go with every choice of women, so the two counts are multiplied.
Men: \(\displaystyle ^7C_3=\frac{7\times6\times5}{3\times2\times1}=35\)
Women: \(\displaystyle ^4C_2=\frac{4\times3}{2}=6\)
Ways \(=35\times6=210\)
Why the other options are incorrect
Option (1) 41 — Incorrect. This is \(35+6\); the two choices go together, so they are multiplied, not added.
Option (2) 462 — Incorrect. This is \(^{11}C_5\), any 5 of the 11 people, without the rule of 3 men and 2 women.
Option (4) 420 — Incorrect. This is \(35\times\,^4P_2\), which wrongly counts the order of the two women.