IGNOU BMTC 103 2026 - Use generating function to prove the identity

Question

Use generating function to prove the identity

$$\sum_{k=0}^n \binom{r}{k} \binom{s}{n - k} = \binom{r + s}{n}.$

20 Jan 2026

Answer :

Word Count : 118

To prove the identity [ \sum_{k=0}^{n} \binom{r}{k} \binom{s}{n-k} = \binom{r+s}{n}, ] we can use generating functions. Consider the generating function for (\binom{r}{k}), which is ((1+x)^r = \sum_{k=0}^{r} \binom{r}{k} x^k), and similarly for (\binom{s}{n-k}), ((1+x)^s = \sum_{k=0}^{s} \binom{s}{k} x^k). Multiplying these two generating functions gives [ (1+x)^r (1+x)^s

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