Question

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

• A. $42$
• B. $43$
• C. $44$
• D. $55$

Answer 1

Answer: (D)

$55$

Answer 2

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 $x^2 = \,^{10}C_1 + \,^{10}C_2 = 55$