Chu-Vandermonde Identity/Proof 3
Theorem
| \(\ds \sum_{k \mathop = 0}^n \binom r k \binom s {n - k}\) | \(=\) | \(\ds \binom r 0 \binom s n + \binom r 1 \binom s {n - 1} + \binom r 2 \binom s {n - 2} + \cdots + \binom r n \binom s 0\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \binom {r + s} n\) |
Informal Proof
The right hand side can be thought of as the number of ways to select $n$ people from among $r$ men and $s$ women.
Each term in the left hand side is the number of ways to choose $k$ of the men and $n - k$ of the women.
$\blacksquare$
Source of Name
This entry was named for Chu Shih-chieh and Alexandre-Théophile Vandermonde.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $\text {I}$