MCQ
The coefficient of ${x^n}$ in $\frac{{{{(1 + x)}^2}}}{{{{(1 - x)}^3}}}$ is
- A$3{n^2} + 2n + 1$
- ✓$2{n^2} + 2n + 1$
- C${n^2} + n + 1$
- D$2{n^2} - 2n + 1$
$ = (1 + 2x + {x^2})[1 + {\,^3}{C_1}x + {\,^4}{C_2}{x^2} + ....$$ + {\,^{.3 + n - 3}}{C_{n - 2}}{x^{n - 2}} + {\,^{3 + n - 2}}{C_{n - 1}}{x^{n - 1}} + {\,^{3 + n - 1}}{C_n}{x^n} + ....]$
Coefficient of
${x^n}$$ = {\,^{3 + n - 1}}{C_n} + {2.^{3 + n - 2}}{C_{n - 1}}{ + ^{3 + n - 3}}{C_{n - 2}}$
$ = {\,^{2 + n}}{C_n} + {2.^{n + 1}}{C_{n - 1}} + {\,^n}{C_{n - 2}}$
$ = \frac{{(2 + n)!}}{{n!2!}} + \frac{{2.(n + 1)!}}{{(n - 1)!2!}} + \frac{{n!}}{{(n - 2)!2!}}$
$ = \frac{{n![(2 + n)(1 + n) + 2(n + 1)n + n(n - 1)]}}{{n!\,2!}}$
$ = 2{n^2} + 2n + 1$.
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