A coach has 11 players and has to pick 5 of them for a relay team. The order of picking does not matter. In how many ways can the 5 players be picked?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 462
Choosing a group where order does not matter is a selection: \(^nC_r=\frac{n!}{r!\,(n-r)!}\).
\(\displaystyle ^{11}C_5=\frac{11\times10\times9\times8\times7}{5\times4\times3\times2\times1}\)
\(\displaystyle =\frac{55440}{120}=462\)
Why the other options are incorrect
Option (1) 55440 — Incorrect. This is \(^{11}P_5\), which counts the order of picking; here order does not matter.
Option (3) 330 — Incorrect. This is \(^{11}C_4\), a choice of 4 players, not 5.
Option (4) 55 — Incorrect. This is \(11\times5\), which is not a count of groups.