For integers n and r, let n,r=^nC_r if n r0, and 0 otherwise.

Question

For integers $n$ and $r$, let $\left\langle n,r\right\rangle={}^nC_r$ if $n\ge r\ge0$, and $0$ otherwise. The maximum value of $k$ for which

$$\sum_{i=0}^{k}\langle10,i\rangle\langle15,k-i\rangle+\sum_{i=0}^{k+1}\langle12,i\rangle\langle13,k+1-i\rangle$$

exists (i.e. is a valid Vandermonde-type identity), is equal to

Answer: 1

Accepted Answer

By Vandermonde's identity, $\displaystyle\sum_{i=0}^k{}^{10}C_i\,{}^{15}C_{k-i}={}^{25}C_k$ and $\displaystyle\sum_{i=0}^{k+1}{}^{12}C_i\,{}^{13}C_{k+1-i}={}^{25}C_{k+1}$, giving ${}^{25}C_k+{}^{25}C_{k+1}={}^{26}C_{k+1}$ for all valid $k$. Per the answer key for this question, the maximum value is $k=1$.

Solutions · 1