There are 16 boys and 16 girls eligible to swim on a mixed relay. In how many ways could the relay team be chosen if the team must contain 2 boys and 2 girls.
Note: the order of the swimmers on the relay matters.
There are 16C2 ways to choose the girls and the boys.
There are 4! ways to arrange the 4 choices. So,
16C2 * 16C2 * 4! = 120^2 * 24 = 345,600 ways
What would the difference be if instead of 16 girls and boys there were 11 or each?
To solve this problem, we can use the concept of combinations.
First, let's choose 2 boys from the 16 available. The number of ways to do this is denoted as "16 choose 2," which can be calculated using the formula:
16! / (2! * (16 - 2)!)
Simplifying this formula, we get:
16! / (2! * 14!)
The exclamation mark represents the factorial of a number, which means multiplying all the positive integers up to that number.
Similarly, we need to choose 2 girls from the remaining 16 available. The number of ways to do this is calculated using the formula:
16! / (2! * 14!)
Now, since the order of the swimmers on the relay matters, we need to multiply the number of possibilities for boys and girls together.
Therefore, the total number of ways the relay team can be chosen is:
(16! / (2! * 14!)) * (16! / (2! * 14!))
Simplifying this expression, we get:
(16! * 16!) / (2! * 2! * 14! * 14!)
Now, we can calculate this expression to find the answer:
(16 * 15 * 14 * 13 * ... * 2 * 1) * (16 * 15 * 14 * 13 * ... * 2 * 1) / (2 * 1 * 2 * 1)
Calculating this expression, we find that the number of ways the relay team can be chosen is 207,900.