A selection is unchanged when its members are written in another order. For n distinct available objects and 0 <= r <= n, nr= n!r!(n-r)!. The ordered count lists each group r! times; dividing by r! removes those duplicate lists. No role or seat is assigned. Identify the available pool and group size. Write one combination, cancel factorials, and state what one outcome represents. Choosing the omitted objects gives nr= nn-r. Example: a team of three from five people has 53=10 possibilities, the same as choosing its two non-members. There is one choice of everyone and one empty choice: 55= 50=1. Check: after counting a committee with combinations, multiplying by r! would incorrectly treat different lists of the same committee as different outcomes.

A selection is unchanged when its members are written in another order. For \(n\) distinct available objects and \(0\le r\le n\), \(\binom nr=\dfrac{n!}{r!(n-r)!}\).

The ordered count lists each group \(r!\) times; dividing by \(r!\) removes those duplicate lists. No role or seat is assigned.

Identify the available pool and group size. Write one combination, cancel factorials, and state what one outcome represents. Choosing the omitted objects gives \(\binom nr=\binom n{n-r}\).

Example: a team of three from five people has \(\binom53=10\) possibilities, the same as choosing its two non-members. There is one choice of everyone and one empty choice: \(\binom55=\binom50=1\).

Check: after counting a committee with combinations, multiplying by \(r!\) would incorrectly treat different lists of the same committee as different outcomes.

Choosing an unordered group

Choosing an unordered group