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7.Binomial Theorem
hard
The coefficient of ${x^4}$ in the expansion of ${(1 + x + {x^2} + {x^3})^n}$ is
A
$^n{C_4}$
B
$^n{C_4}{ + ^n}{C_2}$
C
$^n{C_4} + {\,^n}{C_2} + \,{\,^n}{C_4}{.^n}{C_2}$
D
$^n{C_4} + {\,^n}{C_2} + {\,^n}{C_1}.{\,^n}{C_2}$
Solution
(d) ${(1 + x + {x^2} + {x^3})^n} = \left\{ {{{(1 + x)}^n}{{(1 + {x^2})}^n}} \right\}$
$ = (1 + {\,^n}{C_1}x + {\,^n}{C_2}{x^2} + …. + {\,^n}{C_n}{x^n})$
$(1 + {\,^n}{C_1}{x^2} + {\,^n}{C_2}{x^4} + …. + {\,^n}{C_n}{x^{2n}})$
Therefore the coefficient of $x^4$ = $^n{C_2} + {\,^n}{C_2}.{\,^n}{C_1} + {\,^n}{C_4}$= $^n{C_4} + {\,^n}{C_2} + {\,^n}{C_1}.{\,^n}{C_2}$
Standard 11
Mathematics
