A combination is the number of ways a given number of objects can be selected from a group when the order does not matter (unlike a permutation, in which order does matter). If k objects are selected from a group of n members, the formula for the combination (which can be read as "n choose k") is

${\displaystyle {n\choose k}=\frac{n!}{k!(n-k)!}}$

Combinations are important in probability as well as to the binomial theorem. All the possible combinations of an integer n make up the nth row of Pascal's triangle.

## Formulas

${\displaystyle {n \choose k} = {n \choose (n-k)}}$ Since picking which to use is the same as picking which not to use.

${\displaystyle {n \choose 0} = 1}$ Only 1 option, none of them.

${\displaystyle {n \choose 1} = n}$ Since you pick 1 of them.

${\displaystyle {a \choose 2} = \frac{a^2+a}{2}}$

## Example

How many different groups of three can be chosen from five people?

Since order is not important, we can use the formula for a combination.

${\displaystyle {5 \choose 3} = \frac{5!}{3! (5-3)!} = \frac{5!}{3! \cdot (2)!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1 \cdot (2 \cdot 1)} = \frac{5 \cdot 4}{2} = 10}$

## Practice

Henry wants to choose 3 out of his 7 friends, how many ways can he do this?

Joe wants to choose 4 out of his 6 friends to go fishing with him. However, Jimmy and Andrew do not get along so they cannot be chosen together. How many ways can he do this?