 in combinatorics, vandermonde's identity, or vandermonde's convolution, named after alexandre-thÃ©ophile vandermonde (1772), states that :{m+n \choose r}=\sum_{k=0}^r{m \choose k}{n \choose r-k},\qquad m,n,r\in\mathbb{n}_0, for binomial coefficients. this identity was given already in 1303 by the chinese mathematician zhu shijie (chu shi-chieh). see askey 1975, pp. 59&ndash;60 for the history. there is a q-analog to this theorem called the q-vandermonde identity. == algebraic proof == in general, the product of two polynomials with degrees m and n, respectively, is given by :\biggl(\sum_{i=0}^m a_ix^i\biggr) \biggl(\sum_{j=0}^n b_jx^j\biggr) = \sum_{r=0}^{m+n}\biggl(\sum_{k=0}^r a_k b_{r-k}\biggr) x^r, where we use the convention that ai&nbsp;=&nbsp;0 for all integers i&nbsp;>&nbsp;m and bj&nbsp;=&nbsp;0 for all integers j&nbsp;>&nbsp;n. by the binomial theorem, :(1+x)^{m+n} = \sum_{r=0}^{m+n} {m+n \choose r}x^r. using the binomial theorem also for the exponents m and n, and then the above formula for the product of polynomials, we obtain :\begin{align} \sum_{r=0}^{m+n} {m+n \choose r}x^r &= (1+x)^{m+n}\\ &= (1+x)^m (1+x)^n \\ &= \biggl(\sum_{i=0}^m {m\choose i}x^i\biggr) \biggl(\sum_{j=0}^n {n\choose j}x^j\biggr)\\ &=\sum_{r=0}^{m+n}\biggl(\sum_{k=0}^r {m\choose k} {n\choose r-k}\biggr) x^r, \end{align} where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i&nbsp;>&nbsp;m and j&nbsp;>&nbsp;n, respectively. by comparing coefficients of xr, vandermonde's identity follows for all integers r with 0&nbsp;â¤&nbsp;r&nbsp;â¤&nbsp;m&nbsp;+&nbsp;n. for larger integers r, both sides of vandermonde's identity are zero due to the definition of binomial coefficients. == combinatorial proof == vandermonde's identity also admits a combinatorial double counting proof, as follows. suppose a committee consists of m men and n women. in how many ways can a subcommittee of r members be formed? the answer is :{m+n \choose r}. the answer is also the sum over all possible values of k, of the number of subcommittees consisting of k men and r&nbsp;&minus;&nbsp;k women: :\sum_{k=0}^r{m \choose k}{n \choose r-k}. == geometrical proof == take a rectangular grid of r x (m+n-r) squares. there are : \binom{r+(m+n-r)}{r}=\binom{m+n}{r} paths that start on the bottom left vertex and, moving only upwards or rightwards, end at the top right vertex (this is because r right moves and m+n-r up moves must be made (or vice-versa) in any order, and the total path length
Factorial_and_binomial_topics Mathematical_identities