A basketball team needs 5 players. The team can choose from a group of 7 players. Explain how to find the number of combinations of players that can e on the team.
This is a routine combination question
number of ways = C(7,5) = 7!/(5!2!) = 21
To find the number of combinations of players that can be on the team, you can use the formula for combinations, which is:
nCr = n! / (r!(n-r)!)
Where n is the total number of players to choose from and r is the number of players needed for the team.
In this case, you have 7 players to choose from and you need to select a team of 5 players. So the formula becomes:
7C5 = 7! / (5!(7-5)!)
Now let's calculate step by step:
Step 1: Calculate the factorials:
- 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
- 5! = 5 x 4 x 3 x 2 x 1 = 120
- 2! = 2 x 1 = 2
Step 2: Substitute the factorials into the formula:
7C5 = 5040 / (120 x 2)
Step 3: Simplify the expression:
7C5 = 5040 / 240
Step 4: Calculate the division:
7C5 = 21
Therefore, the number of combinations of players that can be on the team is 21.
To find the number of combinations of players that can be on the basketball team, you can use the concept of combinations in combinatorics. The formula to calculate combinations is given by:
C(n, r) = n! / (r! * (n-r)!)
Where:
- C(n, r) represents the number of combinations of choosing r items from a set of n items.
- n! denotes the factorial of n, which is the product of all positive integers from 1 to n.
- r! denotes the factorial of r.
- (n-r)! denotes the factorial of (n-r).
In this scenario, the team needs to choose 5 players out of 7 available players. Using the combination formula, you can substitute n=7 and r=5 into the formula:
C(7, 5) = 7! / (5! * (7-5)!)
= 7! / (5! * 2!)
Simplifying further:
C(7, 5) = (7 * 6 * 5 * 4 * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (2 * 1))
Here, you notice that most of the terms in the numerator and the denominator cancel out:
C(7, 5) = (7 * 6 * 5 * 4 * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (2 * 1))
= 7 * 6 / (2 * 1)
= 42 / 2
= 21
Therefore, there are 21 different combinations of players that can be on the basketball team.