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 be on the team.
It would be 21. :)
To find the number of combinations of players that can be on the team, you can use the concept of combinations in mathematics.
Step 1: Determine the number of players needed for the team. In this case, it is 5.
Step 2: Identify the total number of players available to be chosen from. In this case, it is 7.
Step 3: Apply the formula for combinations, which is expressed as nCr, where n represents the total number of players available and r represents the number of players needed for the team. The formula for combinations is given by:
nCr = n! / ((n-r)! * r!)
Step 4: Substitute the values into the formula. In this example, we have:
n = 7 (total number of players available)
r = 5 (number of players needed for the team)
Substituting the values into the formula, we have:
7C5 = 7! / ((7-5)! * 5!)
Step 5: Simplify the expression in the formula:
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
(7-5)! = 2! = 2 * 1 = 2
5! = 5 * 4 * 3 * 2 * 1 = 120
So, the simplified expression becomes:
7C5 = 5040 / (2 * 120)
Step 6: Calculate the combination:
7C5 = 5040 / 240
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 team, you can use the concept of combinations. In this case, you have 7 players to choose from, and you need to select 5 players for the team.
To calculate the number of combinations, you can use the formula for combinations:
n C r = n! / (r! * (n - r)!)
Where:
n is the total number of items (players)
r is the number of items to be selected (players in the team)
"!" denotes the factorial operation, which means multiplying a number by all positive integers less than it down to 1.
In this scenario, you need to calculate 7 C 5 (read as "7 choose 5"), which gives you the number of ways to choose 5 players from a group of 7.
Using the combination formula:
7 C 5 = 7! / (5! * (7 - 5)!)
Simplifying further:
7 C 5 = 7! / (5! * 2!)
Now, compute the factorials of 7, 5, and 2:
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
2! = 2 * 1
Substituting these values:
7 C 5 = (7 * 6 * 5 * 4 * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (2 * 1))
Calculating further:
7 C 5 = (7 * 6) / (2 * 1)
7 C 5 = 42 / 2
Finally, you get:
7 C 5 = 21
Therefore, there are 21 different combinations of players that can be on the basketball team.