Why do we divide by r! when calculating combinations?

For selection without repetition, the number of combinations is calculated using the binomial coefficient formula $\binom{n}{r}=n!/[r!(n-r)!]$. This formula counts the number of ways to choose $r$ distinct elements from a set of $n$ elements where order does not matter.

Conditions: The selection is made without repetition.; The elements are distinct.; Order does not matter in the selection.

To calculate the number of combinations with repetition, where order is ignored and repetition is allowed, you use the formula $\binom{n+r-1}{r}$. This counts the number of ways to select $r$ objects from $n$ distinct types.

Conditions: Repetition is allowed.; Order does not matter.; There are $n$ distinct types of objects.; You are selecting exactly $r$ objects.

Reversing a pair creates a different permutation because permutations are ordered arrangements, meaning the sequence of elements matters. However, it does not create a different combination because combinations are unordered selections, where only the membership of the elements in the set matters, not their sequence.

Conditions: The elements being arranged are distinct.; We are comparing ordered arrangements (permutations) with unordered selections (combinations).