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  4. The coefficient of x2 in the expansion of (1 + x + x2 + x3)10 is

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Q. The coefficient of $ x^{2} $ in the expansion of $ (1 + x + x^{2} + x^{3})^{10} $ is

J & K CETJ & K CET 2016Binomial Theorem

Solution:

The given expansion is,
$(1 + x + x^2 + x^3)10 = [1 + x + x^2(1 + x)]^{10}$
$= [(1 + x)(1 + x^2)]^{10} = (1 + x )^{10}(1 + x^2)^{10}$
$=(1 + \,{}^{10}C_1x + \,{}^{10}C_2x^2 + ....+ \,{}^{10}C_{10}x^{10})$
$(1 + \,{}^{10}C_1x^2 + \,{}^{10}C_2x^4 + ....+ \,{}^{10}C_{10} x^{20})$
$\therefore $ Coefficient of $x^2 = \,{}^{10}C_1 + \,{}^{10}C_2 = 55$