# Combinations

Definition: A combination is a grouping of a set of items for which the order does not matter.

For example, ABC would be considered the same combination as BAC or CAB. Note that the order in which the letters are arranged isn’t significant. (When the order does matter, it’s called a permutation.)

When calculating a possible number of combinations, we can use the formula for nCr, where n represents the total number of items to be chosen from, and r represents the number of items we are choosing (“n” different things chosen “r” at a time”):$\dpi{120} \bg_white \large nCr = \frac{nPr}{r!} = \frac{n!}{(n-r)!r!}$

Recall the ! symbol means factorial.

When using combination notation, nCr, we read it as “n choose r”.

Example: How many different ways can a team of 3 be chosen from 5 people?

Solution:

Step 1: First we must decide if the order matters. Here, because it is a TEAM of 3 being chosen, the order in which people are arranged isn’t significant. Therefore, we are being asked to find the number of combinations.

Step 2: Identify the n and r for this question. Remember, combinations are “n different things taken r at a time”. Since we are choosing from 5 different people total, our n is 5. Since we are choosing those people 3 at a time, our r is 3. (“5 different things taken 3 at a time“)

nCr = 5 C 3

Step 3: Plug the values into the formula to determine how many combinations are possible:

$\dpi{120} \bg_white \large \\ {\;\;}nCr = \;\frac{n!}{(n-r)!r!}\\\\ \\5\;\textup{C}\;3 = \;\frac{5!}{(5-3)!3!}\\\\ \\5\;\textup{C}\;3 = \;\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{(2\cdot 1)(3\cdot 2\cdot 1)}\\\\ \\= \frac{120}{12} = 10$

This tells us there are 10 possible 3-person teams that can be formed from 5 total people.