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  4. The coefficient of x7 in the expression (1+x)10+x(1+x)9+x2(1+x)8+...+x10 is :

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Q. The coefficient of $x^7$ in the expression $\left(1+x\right)^{10}+x\left(1+x\right)^{9}+x^{2}\left(1+x\right)^{8}+...+x^{10}$ is :

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Solution:

Coefficient of $x^7$ is
$^{10}C_{7}+^{9}C_{6}+^{8}C_{5}+...+^{4}C_{1}+^{3}C_{0}$
$\underbrace{^4C_0+^4C_1}_{^5C_1}$$+^{5}C_{2}+...+^{10}C_{7}=^{11}C_{7}=330$