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.
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).
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).