Definitions. Let n, k be non-negative integers with n > k. . When n > 0, then the factorial of n (denote n!) is defined as the product of all positive integers less than or equal to n. That is, n!-1-2..-(n-1)-n. By convention, we define . The binomial coefficient n choose k (denote C(n, k) or ,) is given by (20 points) Use mathematical induction to prove that the following equalities and inequalities hold for all positive integers n. n- For this problem, if needed, you may use (without proof) the fact that n n1 for all positive integers n