IGNOU BMTC 103 2026 - For any natural numbers n and k, prove that .

Question

For any natural numbers n and k, prove that $P_n^k = P_{n-1}^{k-1} + P_{n-k}^k$ .

20 Jan 2026

Answer :

Word Count : 321

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 first object) to be in the first position. Then, we need to select the remaining (k-1) objects from the remaining (n-1) objects. The number of ways to do this is (P_{n-1}^{k-1}). 2. Case 2: Do not include that specific object in the permutation. Then all

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