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

Question

The coefficient of x2 in the expansion of (1 + x + x2 + x3)10 is (A)  42  (B)  43  (C)  44  (D)  55

Tardigrade Admin · Accepted Answer

The given expansion is, (1 + x + x2 + x3)10 = [1 + x + x2(1 + x)]10 = [(1 + x)(1 + x2)]10 = (1 + x )10(1 + x2)10 =(1 + 10C1x + 10C2x2 + ....+ 10C10x10) (1 + 10C1x2 + 10C2x4 + ....+ 10C10 x20) ∴ Coefficient of x2 = 10C1 + 10C2 = 55

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

$42$

$43$

$44$

$55$

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

42

43

44

55

The given expansion is, (1+x+x2+x3)10=[1+x+x2(1+x)]10 =[(1+x)(1+x2)]10=(1+x)10(1+x2)10 =(1+10C1​x+10C2​x2+....+10C10​x10) (1+10C1​x2+10C2​x4+....+10C10​x20) ∴ Coefficient of x2=10C1​+10C2​=55

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

$42$

$43$

$44$

$55$