combinations and probability

Question

The coach has to select 2 co-captains from the 15 girls out for soccer. How many sections are possible?

Answer 1

Since you are picking 2 out of the 15 players on a soccer team to be co-captains, the order in which you pick them doesn't matter. To illustrate this concept: Suppose you wanted to pack 2 snacks in your lunch box and there were 3 items to chose from: a banana, a cookie, and an apple. And say that you wanted to mix up things so that you are not eating the same two snacks every day. How many different "combinations" can be made?

Banana & Cookie

Banana & Apple

Cookie & Apple

Notice how Banana & Cookie is the same as Cookie & Banana. Meaning... The order that you pick your 2 snacks does not matter. Banana & Cookie is only one combination. In this simple snack problem, there are 3 different types of combinations possible.

It becomes way more difficult to list all of the possible combinations when dealing with larger numbers of items. Going back to your original problem of 2 co-captains picked from 15 girls. You would need to list every possible combination of 2 girls, being sure to not duplicate and combination. Instead, we will use the combination formula:

There are 105 different combinations of 2 co-captains out of 15 players.

Answer 2

Since co-captains are indistinguishable roles, then order does not matter in the selecting of 2 co-captains. All that matters is which 2 girls are selected out of the 15. This is a combination, notated as ₁₅C₂. The formula is (n!)/[r!(n - r)!]. So this is 15!/(2!13!). Since 13! is in the denominator, it cancels out completely, and in the numerator, everything from 13 down to 1 cancels, leaving only 15 • 14. There is still a 2! remaining in the denominator, which equals 2. So the answer is 15 • 7, which equals 105.

Most scientific calculators have a function for this also, but this is pretty easy to work out without a calculator.