A group of 4 volunteers is to be made of 2 men chosen from 6 men and 2 women chosen from 5 women. In how many ways can the group be made?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 150
The men and the women are chosen separately, and each choice of men can go with each choice of women, so the two counts are multiplied. Order does not matter in a group, so each count is a combination.
2 men out of 6: \(\displaystyle ^6C_2=\frac{6\times5}{2}=15\)
2 women out of 5: \(\displaystyle ^5C_2=\frac{5\times4}{2}=10\)
Groups \(=15\times10=150\)
Why the other options are incorrect
Option (2) 25 — Incorrect. This is \(15+10\); the two choices are made together, so they are multiplied, not added.
Option (3) 210 — Incorrect. This is \(^{11}C_4\), any 4 of the 11 people, which ignores the 2 men and 2 women rule.
Option (4) 300 — Incorrect. This is \(2\times150\); the order of choosing does not matter, so nothing is doubled.