Arrangements A(n,k)

Arrangements A(n,k)

An arrangement is an ordered selection of k elements out of n distinct ones: A(n,k) = n·(n−1)·…·(n−k+1) = n! / (n−k)!. Here order matters, and that's exactly what sets arrangements apart from combinations, which use the same formula divided by k!. Take a 4-digit PIN with no repeated digits: it has A(10,4) = 10·9·8·7 = 5040 possibilities. Picking the podium (1st, 2nd, 3rd) among 8 runners gives A(8,3) = 8·7·6 = 336. A few edge cases worth knowing: A(n,0) = 1 (the empty arrangement), A(n,n) = n! (a full permutation), and A(n,k) = 0 when k > n.

Applications and context

It shows up in PIN/password design when you count the keyspace, in podium selection and ranking, in shift scheduling where positions are distinct, in elections (president, vice, secretary out of an assembly), and in the combinatorics problems on competitive exams like ENEM, OBM, OBMEP and the Brazilian vestibulares.

FAQ

Arrangement vs. combination? An arrangement cares about order, so 1st/2nd/3rd count as different outcomes. A combination doesn't, so you just get "the three winners".

Arrangement vs. permutation? A permutation uses all n elements, while an arrangement picks only k of them.

With repetition allowed? Then the count becomes nᵏ instead. A 4-digit PIN that allows repeats, for instance, has 10⁴ = 10000 options.