A class contains 27 boys and 29 girls. In how many ways can you select a team of four people from the class such that every member of the team is of the same sex?
We’ll need to know how many ways we can get a group of four boys, and we’ll add that value to the number of ways we’ll get a group of four girls. In a four-person team, order doesn’t matter, and that means this is a combination.
The formula for this is a combination of 𝑛 objects, where we’re choosing 𝑟, number of them, equals 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. The combination of boys is 27 boys, where we’re choosing four, four factorial times 27 minus four factorial. 27 minus four equals 23.
We need to follow this same setup for the girls. We’re looking for the combination of 29 girls, taking four at a time. It’ll be 29 factorial over four factorial times 29 minus four factorial. 29 minus four is 25.
If we bring down our equations, we can start to simplify them. Remember what factorial means. 27 factorial is equal to 27 times 26 times 25 times 24, all the way down to one. We could also say that 27 factorial is equal to 27 times 26 times 25 times 24 times 23 factorial.
The reason we write it like this is to notice that something in the numerator and the denominator cancels out. There’s a 23 factorial in the numerator and the denominator. After those cancel out, we’re left with 27 times 26 times 25 times 24 over four factorial, which can be rewritten as four times three times two times one.
We’ll do some more simplifying. 24 divided by two equals 12, 27 divided by three equals nine, and 12 divided by four equals three. To find the combinations, we’ll have to multiply nine times 26 times 25 times three. It equals 17550.
Now we wanna simplify the combination of girls. We’ll take our 29 factorial and rewrite it as 29 times 28 times 27 times 26 times 25 factorial. And then the 25 factorial in the numerator and the denominator cancel out. We need to expand the four factorial: four times three times two times one.
We can then simplify. 28 divided by four equals seven, 27 divided by three equals nine, and 26 divided by two equals 13. To find the combination of girls, multiply 29 times seven times nine times 13, which equals 23751.
To find the total way we could select a team of four people from the class, we need to add the boy teams and the girl teams. Adding them together, we get 41301. There are 41301 ways we can select a team of four people from the class such that every member of the team is of the same sex.