JEE Advance - Mathematics (1989 - No. 15)
Using mathematical induction, prove that $${}^m{C_0}{}^n{C_k} + {}^m{C_1}{}^n{C_{k - 1}}\,\,\, + .....{}^m{C_k}{}^n{C_0} = {}^{\left( {m + n} \right)}{C_k},$$
where $$m,\,n,\,k$$ are positive integers, and $${}^p{C_q} = 0$$ for $$p < q.$$
where $$m,\,n,\,k$$ are positive integers, and $${}^p{C_q} = 0$$ for $$p < q.$$
Base Case: Verify the equation holds for a small value of k (e.g., k=0 or k=1).
Inductive Hypothesis: Assume the equation holds for some arbitrary k=l, where l is a positive integer.
Inductive Step: Prove that the equation holds for k=l+1, assuming it holds for k=l.
Using the identity $${}^n{C_r} + {}^n{C_{r-1}} = {}^{n+1}{C_r}$$ to show the equation holds
All of the above are important steps in the mathematical induction
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