First slide

Multinomial expansion

Question

The coefficient of x⁴ in the expansion of ${(1 + x + x^{2} + x^{3})}^{11}$ is

NA

A

900
B

909
C

990
D

999

Solution

Let $E = {(1 + x + x^{2} + x^{3})}^{11}$
$\begin{aligned} = {[(1 + x) (1 + x^{2})]}^{11} \\ = (1 + x)^{11} {(1 + x^{2})}^{11} \end{aligned}$
General term of E is,
$T (r, k) = [^{11} C_{r} x^{r}) (^{11} C_{k} {(x^{2})}^{k}]$
$=^{11} C_{r} \cdot^{11} C_{k} x^{r + 2 k}$
For taking coefficient of x⁴ take
r+2k=4
r 0 1 2 3 4 5
$k = \frac{4 - r}{2}$ 2 x 1 x 0 x
. Coefficient of x⁴
$\begin{array}{l} = T_{(0, 2)} + T_{(2, 1)} + T_{(4, 0)} \\ =^{11} {C_{0}}^{11} C_{2} +^{11} {C_{2}}^{11} C_{1} +^{11} {C_{4}}^{11} C_{0} \\ = 55 + 55 \times 11 + 330 \\ = 55 + 605 + 330 = 990 \end{array}$

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