The sum of coefficients of odd power of $x$ in the expansion of $(1 + 2 x + 3 x^{2} + 4 x^{3})^{4}$ is

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Solution

$(1 + 2 x + 3 x^{2} + 4 x^{3})^{4} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{12} x^{12}$
Putting $x = 1$ , we get
$a_{0} + a_{1} + a_{2} + . . . + a_{12} = 10^{4} . . . (1)$

Putting $x = - 1$ , we get
$a_{0} - a_{1} + a_{2} - . . . + a_{12} = (- 2)^{4} . . . (2)$

Adding eqn(1) and (2), we get
$2 (a_{0} + a_{2} + a_{4} + . . . + a_{12}) = 10^{4} + 2^{4}$
$\Rightarrow a_{0} + a_{2} + a_{4} + . . . + a_{12} = 5008$

$∴ a_{1} + a_{3} + a_{5} + . . . + a_{11} = 10^{4} - 5008 = 4992$

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