cho khai triển
$\left ( 1+x+x^{2}+...+x^{14} \right )^{15}=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{210}x^{210}$
chứng minh rằng:
$C^{0}_{15}a_{15}-C^{1}_{15}a_{14}+C^{2}_{15}a_{13}-...-C^{15}_{15}a_{0}=-15$
Xét khai triển $(1-x^{15})^{15}=(1-x)^{15}(1+x+x^{2}+...+x^{14})^{15}$
$=\sum_{k=0}^{15}C^{k}_{15}(-x)^{k}(a_{0}+a_{1}x+a_{2}x^{2}+....+a_{210}x^{210})$
$+(1-x^{11})^{11}=\sum_{i=0}^{15}(-x)^{i} $

Đồng nhất hệ số của $x^{15}$ trong 2 khai triển trên, ta được:

$-C^{1}_{15}=-C^{15}_{15}a_{0}+C^{14}_{15}a_{2}-.....+C^{0}_{15}a_{15}$
$\Leftrightarrow đpcm$

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