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  4. In the expansion of (1 + x + x2 + x3)10 , the coefficient of x7 is

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Q. In the expansion of $\left(1 + x + x^{2} + x^{3}\right)^{10}$ , the coefficient of $x^{7}$ is

NTA AbhyasNTA Abhyas 2022

Solution:

$ \left(1+x+x^{2}+x^{3}\right)^{10}=\left(1+x\left(1+x^{2}\right)^{10}\right. $
$\begin{array}{cccccc}(1+x)^{10} & \left(1+x^{2}\right)^{10} & \text { coefficient } & \text { from }(1+x)^{10} & \text { coefficient } & \text { from }\left(1+x^{2}\right)^{10} \\ 7 & 0 & { }^{10} C_{7} & { }^{10} C_{0} \\ 5 & 2 & { }^{10} C_{5} & { }^{10} C_{1} \\ 3 & 4 & { }^{10} C_{3} & { }^{10} C_{2} \\ 1 & 6 & { }^{10} C_{1} & { }^{10} C_{3}\end{array}$
Coefficient of $x^{7}=120 \times 1+252 \times 10+120 \times 45+10 \times 120=9240$