The coefficient of xr in the expansion of 1-4x 2 -x 4 1-x 4

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Expansions of the Form (1+x)^(-n) and (1-x)^(-n)

The coefficie...

Question

The coefficient of $x^{r}$ in the expansion of $\frac{1 + 4 x^{2} + x^{4}}{{(1 - x)}^{4}}$

r^{2} + r

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r^{3} + r

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r^{3} + 3 r

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r^{3} + 2 r

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Solution

The correct option is C $r^{3} + 3 r$
$(1 + 4 x^{2} + x^{4})$ ${(1 - x)}^{- 4}$
$(1 + 4 x^{2} + x^{4})$ $(. . . . . . . . . . . . . . . . . . . . +^{4 + r - 1} C r x^{r} + . . . . +^{4 + r - 2 - 1} C_{r - 2} . x^{r - 2} . . . +^{4 + r - 4 - 1} C_{r - 4} . x^{r - 4} . . . . . . . . . . . . .)$
$(1 + 4 x^{2} + x^{4})$ $(. . . . . . . . . . . . . . . . . . . . +^{r + 3} C r x^{r} + . . . . +^{r + 1} C_{r - 2} . x^{r - 2} . . . +^{r - 1} C_{r - 4} . x^{r - 4} . . . . . . . . . . . . .)$
$(. . . . . . . . . . . (^{r + 3} C_{r} + 4^{r + 1} C r - 2 +^{r - 1} C r - 4) x^{r} . . . . . . . . . .)$
$(. . . . . . . . . . . (^{r + 3} C_{r} + 4^{r + 1} C r - 2 +^{r - 1} C r - 4) x^{r} . . . . . . . . . .)$
Hence, the coefficient of $x^{r}$ is
$\frac{(r + 3) . (r + 2) . (r + 1)}{6}$ $+$ $4 \frac{(r + 1) . (r) (r - 1)}{6}$ $+$ $\frac{(r - 1) (r - 2) (r - 3)}{6}$
$=$ $\frac{r^{3} + 6 r^{2} + 11 r + 6}{6}$ $+ 4$ $(\frac{r^{3} - r}{6})$ $+$ $\frac{r^{3} - 6 r^{2} + 11 r - 6}{6}$
$=$ $\frac{6 r^{3} + 22 r - 4 r}{6}$
$=$ $\frac{6 r^{3} + 18 r}{6}$
$=$ $r^{3} + 3 r$

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Similar questions

if $(1 + x + \frac{1}{x})^{4} = \sum_{r_{1} + r_{2} + r_{3}} = 4 \frac{4!}{r_{1}! \times r_{2}! \times r_{3}!} \times (1)^{r_{1}} (x)^{r_{2}} (\frac{1}{x})^{r_{3}}$ how many values can $r_{1}$ take so that the terms in the expansion will be independent of x.