+274 DO NOT POST THE SOLUTION/ANSWER IF YOU ARE UNSURE ABOUT IT.
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?
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.
+274 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
\(\begin{align*} \frac{12\times11\times10\times9}{4\times3\times2}\times\frac{8\times7\times6\times5}{4\times3\times2} &= 11\times 10 \times 9 \times 7 \times 5 \\ &= 99\times 35\times 10 \\ &= (3500-35)\times 10 \\ &= \boxed{34,650\text{ ways}}. \end{align*}\)
