A school debate club has 9 members. The teacher has to choose 4 of them to represent the school at a contest. In how many ways can the 4 members be chosen?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 126
Only which members go matters, not the order in which they are picked, so this is a selection (combination): \(^nC_r=\frac{n!}{r!\,(n-r)!}\).
4 members out of 9: \(\displaystyle ^9C_4=\frac{9\times8\times7\times6}{4\times3\times2\times1}\)
\(\displaystyle =\frac{3024}{24}=126\)
Why the other options are incorrect
Option (1) 3024 — Incorrect. This is \(^9P_4\), which counts the same group of 4 again for each of its \(4!=24\) orders.
Option (2) 36 — Incorrect. This is \(^9C_2\), the number of ways to choose only 2 members.
Option (3) 84 — Incorrect. This is \(^9C_3\), the number of ways to choose only 3 members.