Problem 5. Let a1, a2, as, a4, and as be any points in the set [0, 1] x [0, 1]. Prove that at least two of these points are within distance , of each other. Hint: Use the Pigeonhole Principle. Problem 6. Prove the Binomial Theorem: For any nonnegative integer n, Hint: Use induction. Problem 7. Let P(n, k) denote the number of k-permutations of an n element set (in other words, P(n, k) is the number of non-repeating ordered lists of length k whose entries come from an n element set). Prove that for all n e Z30 and for all 0