If ^20C_r is the coefficient of x^r in the expansion of (1+x)^20, then the value of _r=0^2…

Question

If ${}^{20}C_r$ is the coefficient of $x^r$ in the expansion of $(1+x)^{20}$, then the value of $\displaystyle\sum_{r=0}^{20}r^2\,{}^{20}C_r$ is equal to

Accepted Answer

Using $r\cdot{}^nC_r=n\cdot{}^{n-1}C_{r-1}$ and $r(r-1)\cdot{}^nC_r=n(n-1)\cdot{}^{n-2}C_{r-2}$, we get $r^2\,{}^nC_r=n(n-1)\,{}^{n-2}C_{r-2}+n\,{}^{n-1}C_{r-1}$.

With $n=20$: $$\sum_{r=0}^{20}r^2\,{}^{20}C_r = (20\times19)\sum{}^{18}C_r+20\sum{}^{19}C_r = 380\times2^{18}+20\times2^{19}=420\times2^{18}$$

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