The value of _r=0^6(^6C_r^6C_6-r) is equal to

Question

The value of $\displaystyle\sum_{r=0}^{6}\left({}^6C_r\cdot{}^6C_{6-r}\right)$ is equal to

Accepted Answer

Comparing the coefficient of $x^6$ in $(1+x)^6(1+x)^6=(1+x)^{12}$:

$${}^6C_0\,{}^6C_6+{}^6C_1\,{}^6C_5+\cdots+{}^6C_6\,{}^6C_0={}^{12}C_6=924$$

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