10 boys and 10 girls are eligible to swim in 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 10C2 ways to choose the girls and the boys.
There are 4! ways to arrange the 4 choices. So,
10C2 * 10C2 * 4! = 90^2 * 24 = 194,400 ways
first choose which two girls will race (no order yet)
combinations of 10 girls 2 at a time = 10!/[ 2! (8!) ] = 45
same for the boys 45 ways
for each choice of two girls I can chose a choice of two boys
so if I choose group 1 of boys, I can choose any of the 45 groups of girls
if I choose group 2 of two boys , I can choose any of the 45 groups of girls
etc
45*45 ways
now I have four people in a room, how many ways can I order them
4 * 3 * 2 * 1 =4! = 24
so I end up with 45 * 45 * 24
I do not want to be the coach
To solve this problem, we can use combination and permutation principles.
First, let's choose 2 boys out of the 10 available. We can do this by using combination notation (denoted by "C"). The number of ways to choose 2 boys out of 10 is C(10, 2).
Next, let's choose 2 girls out of the remaining 10 available. Again, we can use combination notation to find the number of ways to choose 2 girls out of 10, which is C(10, 2).
Once we have selected the 2 boys and 2 girls, we need to assign them to specific positions on the relay team. Since the order matters, we need to use permutation notation (denoted by "P").
For the first position on the relay team, we have 4 options (2 boys and 2 girls). We can use permutation notation to describe this, which is P(4, 1).
For the second position, we have 3 remaining candidates (3 boys or 3 girls). Again, we use P(3, 1).
For the third position, we have 2 remaining candidates. We use P(2, 1).
Finally, for the last position, only 1 candidate is left. We use P(1, 1).
To calculate the total number of ways, we multiply all these values together. Hence, the total number of ways to form the relay team with 2 boys and 2 girls, where the order matters, is:
C(10, 2) * C(10, 2) * P(4, 1) * P(3, 1) * P(2, 1) * P(1, 1)
Simplifying this expression, we get:
(10!)/(2! * 8!) * (10!)/(2! * 8!) * 4! * 3! * 2! * 1!
Now, we can calculate the value.