Word Count : 321

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We need to prove that for any natural numbers (n) and (k), [ P_n^k = P_{n-1}^{k-1} + P_{n-k}^k, ] where (P_n^k) denotes the number of permutations of (n) objects taken (k) at a time. By definition, [ P_n^k = \frac{n!}{(n-k)!}. ] Consider a set of (n) distinct objects. To form a permutation of (k) objects: 1. Case 1: Choose a specific object (say the ______ _______ _____ ________ _________ _________ _____ ________ __________ __________.
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