Sum over k of r-k Choose m by s+k Choose n
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Theorem
Let $m, n, r, s \in \Z_{\ge 0}$ such that $n \ge s$.
Then:
- $\ds \sum_{k \mathop = 0}^r \binom {r - k} m \binom {s + k} n = \binom {r + s + 1} {m + n + 1}$
where $\dbinom {r - k} m$ etc. are binomial coefficients.
Proof
\(\ds \) | \(\) | \(\ds \sum_{k \mathop = 0}^r \binom {r - k} m \binom {s + k} n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^r \binom {-\left({m + 1}\right)} {r - k - m} \binom {-\left({n + 1}\right)} {s + k - m} \left({-1}\right)^{r - m + s - m}\) | Moving Top Index to Bottom in Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \binom {-\left({m + 1}\right) - \left({n + 1}\right)} {r - m + s - m} \left({-1}\right)^{r - m + s - m}\) | Chu-Vandermonde Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \binom {r + s + 1} {m + n + 1}\) | Moving Top Index to Bottom in Binomial Coefficient |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $\text {I} \ (25)$