In how many ways can a group of 6 boys and 2 girls be chosen from 7 boys and 3 girls?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 21
The number of ways of choosing \(r\) things out of \(n\) is \(^nC_r\). The boys and the girls are chosen independently, so the two counts are multiplied.
Ways to choose 6 boys out of 7: \(^7C_6=7\)
Ways to choose 2 girls out of 3: \(^3C_2=3\)
Total \(=7\times3=21\)
Why the other options are incorrect
Option (1) 50 — Incorrect. It does not come from \(^7C_6\) and \(^3C_2\); the count is only 21.
Option (2) 120 — Incorrect. This is \(^{10}C_3\) (equal to \(^{10}C_7\)), a count that ignores the 6 boys and 2 girls split.
Option (4) 20 — Incorrect. The two choices are multiplied, \(7\times3=21\), not 20.