Combinations

A combination is a selection of objects where order does not matter.

The number of combinations of \( n \) objects taken \( r \) at a time is \( {n \choose r} = \frac{n!}{r!(n-r)!} \) (

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

). The term ${n \choose r}$ is called the binomial coefficient and is read as "$n$ choose $r$".

Examples:

• There are \( {5 \choose 2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10 \) ways to choose $2$ students from $5$ students.
• There are \( {7 \choose 3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!} = \frac{5040}{6 \cdot 24} = \frac{5040}{144} = 35 \) ways to form a committee of $3$ people from $7$ candidates.

Combinations are used when the order of selection does not affect the result.

Choosing people for a team usually involves combinations, not permutations.