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juliie Let m ≥ n and let A = {1,2,...,m} and let B = {1,2,...,n}. In this exercise you will determine a formula for the number of surjective functions from A to B. For each i ∈ B, let Ai denote the set of all functions f : A → B such that i is not in the image of f, that is, for all x ∈ A we have f(x) 6= i. Thus the set of all functions from A to B which are not surjective is A1 ∪ A2 ∪ ··· ∪ An.
1. a) Let 1 ≤ k ≤ n. In how many ways can we choose integers i1,i2,...,ik such that 1 ≤ i1 < i2 < ··· < ik ≤ n?
b) Let 1 ≤ i1 < i2 < ··· < ik ≤ n. Show that |Ai1 ∩ Ai2∩ ··· ∩ Aik| = (n − k)^m