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Thomas, Carrie and Lenny each captain a different one of three hockey teams. Each captain will choose four players from a pool of 12 players, with each player chosen for only one team. How many different ways can the teams be formed?

supremecheetah May 22, 2023

#1**0 **

Thomas can choose from 12 players to form his team, Carrie can choose from the 8 remaining players, and Lenny can choose from the 5 remaining players. Therefore, there are 12×8×5=480 ways to form the teams.

Guest May 22, 2023

#4**0 **

Correct! Here's the full and detailed solution. Let me know if you have any questions!

Thomas can choose any \(4\) out of \(12\) players, which is \(\binom{12}{4}\) distinct possibilities for a team. Carrie can choose any \(4\) out of the remaining \(8\) players, which is \(\binom{8}{4}\) distinct possibilities for a team. Lenny has only \(1\) choice for his team, whichever \(4\) have not yet been chosen. Combining all this yields

\(\begin{align*} \frac{12\times11\times10\times9}{4\times3\times2}\times\frac{8\times7\times6\times5}{4\times3\times2} &= 11\times 10 \times 9 \times 7 \times 5 \\ &= 99\times 35\times 10 \\ &= (3500-35)\times 10 \\ &= \boxed{34,650\text{ ways}}. \end{align*}\)

supremecheetah May 22, 2023