In hockey, 6 players from each team are on the ice at a time: 1 goalie, 2 defence men, and 3 forwards.
A school hockey team has a total of 17 players: 2 goalies, 6 defensive players, and 9 forwards.
A) How many of these starting line ups include Sam, one of the team's goalies?
B) if the 2 goalies on the team are sick and cannot play, the coach will need to pick one of the defensive players to put in for the goalie. How many possible line ups are there now?
*****The answers to these questions should have something with combinations.
Please help. thanks.
a) since the goalie is picked, you still need 2 out of the 6 defensemen --> C(6,2)
and then 3 of the 9 forwards ----> C(9,3)
so no. of lineups = 1xC(6,2)xC(9,3)
= (15)(84) = 1260
b) Choose a goalie from the 6 defensemen
= C(6,1)
now choose 3 defensemen from the reamining 5
= C5,3)
and the 3 forwards from 9
= C(9,3)
so 6(10)(84) = 5040
oh ok, that makes sense. thanks
To answer these questions using combinations, we need to understand the concept of combinations and how to apply it in these scenarios.
A) How many of these starting line ups include Sam, one of the team's goalies?
To calculate the number of lineups that include Sam as the goalie, we need to select the remaining players from the available options while keeping the positions in mind. Let's break it down step by step:
1. Select Sam as the goalie: Since Sam is a fixed position, there is only one way to select Sam.
2. Select 2 defensemen: From the remaining 16 players (excluding Sam), we need to choose 2 defensemen. To calculate the number of ways to do this, we use the combination formula:
C(n, r) = n! / (r! * (n - r)!)
Where:
n = total number of players to choose from
r = number of defensemen to select
In this case, we have 16 players to choose from (excluding Sam) and need to select 2 defensemen. So the calculation would be:
C(16, 2) = 16! / (2! * (16 - 2)!).
Solving this equation will give us the number of ways to select 2 defensemen from 16 players.
3. Select 3 forwards: From the remaining players (after selecting Sam and the defensemen), we need to choose 3 forwards. Again, using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, we have 14 players left to choose from (since 2 defensemen and Sam are already selected), and we need to select 3 forwards. So the calculation would be:
C(14, 3) = 14! / (3! * (14 - 3)!).
Solving this equation will give us the number of ways to select 3 forwards from the remaining 14 players.
4. Calculate the total number of lineups: Now, to get the total number of lineups including Sam as the goalie, we multiply the results from the previous steps:
Total number of lineups = 1 * C(16, 2) * C(14, 3).
Solve this equation to find the answer.
B) If the 2 goalies on the team are sick and cannot play, and the coach needs to pick one of the defensive players to play as the goalie, we have a different scenario.
1. Select 1 player from the defensive players: Since the coach needs to choose one defensive player to play as the goalie, we have 6 options available.
2. Select 2 defensemen: From the remaining 15 players (excluding the chosen defensive player/goalie), we select 2 defensemen using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, we have 15 players left to choose from (after selecting the goalie), and we need to select 2 defensemen. So the calculation would be:
C(15, 2) = 15! / (2! * (15 - 2)!).
Solve this equation to find the number of ways to select 2 defensemen from the remaining players.
3. Select 3 forwards: From the remaining players (after selecting the goalie and the defensemen), our goal is to select 3 forwards. Using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, we have 12 players left to choose from (after selecting the goalie and defensemen), and we need to select 3 forwards. So the calculation would be:
C(12, 3) = 12! / (3! * (12 - 3)!).
Solve this equation to find the number of ways to select 3 forwards from the remaining players.
4. Calculate the total number of lineups: Now, to get the total number of possible lineups, we multiply the results from the previous steps:
Total number of lineups = (number of goalie selections) * (number of defensemen selections) * (number of forward selections).
Solve this equation to get the answer.
Remember to perform the necessary calculations based on the combination formulas provided above to find the exact number of lineups.