When is an arrangement a combination, and when a permutation?

An arrangement is a permutation when we care about the order of the elements in it, and a combination when we do not. So if we are for example choosing a group of 3 people from 10 students, we do not care about the order; we want a combination and hence use n!/(r!(n-r)!) = 10!/(3!7!). If we now decided that when we choose this group, the first person chosen is the team leader, the second is secretary, and the third is treasurer, then there are more ways of assembling different groups as we care about the distribution of roles; the order in which students are picked matters. Hence we now use the rule for permutations, n!/(n-r)! = 10!/7!.