Question

There are 20 members of a basketball team.

(a) The coach must select 12 players to travel to an away game. How many ways are there to select the players who will travel?
(b) From the 12 players who will travel, the coach must select her starting line-up. She will select a player for each of the five positions: center, power forward, small forward, shooting guard and point guard. How many ways are there for her to select the starting line-up?
(c) From the 12 players who will travel, the coach must select her starting line-up. She will select a player for each of the five positions: center, power forward, small forward, shooting guard and point guard. However, there are only three of the 12 players who can play center. Otherwise, there are no restrictions. How many ways are there for her to select the starting line-up?

Answer 1

There are 125,970 ways to select the players who will travel to an away game. There are 79,833,600 ways to select the starting line-up. With a restriction on the center position, there are 12 ways to select the starting line-up.

Step-by-step explanation:

(a) The coach must select 12 players to travel to an away game:

There are 20 members on the basketball team, and the coach needs to select 12 players to travel. This is a combination problem, since the order doesn't matter. The formula for calculating combinations is: nCr = n! / (r! * (n-r)!), where n is the total number of players and r is the number of players to be selected.

Using this formula, we can calculate the number of ways to select the traveling players: 20C12 = 20! / (12! * (20-12)!) = 125,970 ways.

(b) The coach must select her starting line-up:

There are 12 players who will travel, and the coach needs to select 5 players for the starting line-up. This is a permutation problem, since the order does matter. The formula for calculating permutations is: nPr = n! / (n-r)!, where n is the total number of players and r is the number of players to be selected.

Using this formula, we can calculate the number of ways to select the starting line-up: 12P5 = 12! / (12-5)! = 12! / 7! = 79,833,600 ways.

(c) The coach must select her starting line-up, with a restriction on the center position:

There are 12 players who will travel, and only 3 players who can play center. The remaining 4 positions have no restrictions. We can calculate the number of ways to select the starting line-up by first selecting the center player (3 ways), and then selecting the players for the other positions (4P4 = 4 ways).

Therefore, the total number of ways to select the starting line-up with the restriction on the center position is: 3 * 4 = 12 ways.