Assume that there are 11 girls and 9 boys in the neighborhood club, and a team of 6 is to be selected.How many different teams can be selected if each team must contain exactly 3 girls and 3 boys?
To solve this problem, we need to find the number of ways to choose 3 girls out of 11 and 3 boys out of 9.
The number of ways to choose 3 girls out of 11 can be calculated using the combination formula, also known as "nCr".
The formula for calculating the number of combinations is nCr = n! / (r!(n-r)!), where n is the total number of objects, r is the number of objects to be chosen, and ! denotes the factorial operation.
In this case, we want to calculate 11C3, which is equal to 11! / (3!(11-3)!):
11C3 = 11! / (3! * 8!)
To calculate the factorial of a number, you multiply that number by all the positive integers less than it. For example, 3! = 3 * 2 * 1 = 6.
Using this logic, we can simplify the equation:
11C3 = 11! / (3! * 8!)
= (11 * 10 * 9 * 8!) / (3! * 8!)
= (11 * 10 * 9) / 3!
Canceling out the terms, we get:
11C3 = (11 * 10 * 9) / (3 * 2 * 1)
= 165
So, there are 165 different ways to choose 3 girls out of 11.
Similarly, we want to calculate 9C3, which is equal to 9! / (3!(9-3)!):
9C3 = 9! / (3! * 6!)
Using the same process, we simplify the equation:
9C3 = (9 * 8 * 7 * 6!) / (3! * 6!)
= (9 * 8 * 7) / 3!
Canceling out the terms, we get:
9C3 = (9 * 8 * 7) / (3 * 2 * 1)
= 84
So, there are 84 different ways to choose 3 boys out of 9.
To find the total number of different teams that can be selected, we multiply the number of ways to choose 3 girls by the number of ways to choose 3 boys:
Total teams = 165 * 84
= 13860
Therefore, there are 13,860 different teams that can be selected if each team must contain exactly 3 girls and 3 boys.