The coefficient of x4 in the expansion of 1+x+x2+x311 is

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

$\begin{array}{l} = T_{(0, 2)} + T_{(2, 1)} + T_{(4, 0)} \\ =^{11} {C_{0}}^{11} C_{2} +^{11} {C_{2}}^{11} C_{1} +^{11} {C_{4}}^{11} C_{0} \\ = 55 + 55 \times 11 + 330 \\ = 55 + 605 + 330 = 990 \end{array}$

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

Solution:

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