How do you calculate the number of combinations with repetition when choosing r objects from n types?

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.

We divide by $r!$ because the permutation count includes all possible internal orderings of each selected group. Since combinations are unordered selections, these internal orderings are redundant.

Conditions: The selection is made without repetition.; Order does not matter in the final count.

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).