The value of r for which ^20C_r,^20C_0+^20C_r-1,^20C_1+^20C_r-2,^20C_2++^20C_0,^20C_r is m…

Question

The value of $r$ for which ${}^{20}C_r\,{}^{20}C_0+{}^{20}C_{r-1}\,{}^{20}C_1+{}^{20}C_{r-2}\,{}^{20}C_2+\cdots+{}^{20}C_0\,{}^{20}C_r$ is maximum, is:

Accepted Answer

This sum equals $\left({}^{20}C_0\right)^2+\left({}^{20}C_1\right)^2+\cdots$ when $r=20$ (giving $\displaystyle\sum\left({}^{20}C_k\right)^2={}^{40}C_{20}$), which is the maximum possible value of this Vandermonde-type sum.

$r=20$

Solutions · 1