If k is the coefficient of x 4 in the expansion of ( 1 + 2 x + 3 x 2 ) 6 , then the sum of the digits of k is

Question

If $k$ is the coefficient of $x^{4}$ in the expansion of ${(1 + 2 x + \frac{3}{x^{2}})}^{6}$ , then the sum of the digits of $k$ is _______.

Smart Achievers · Accepted Answer

(6)

$The general term is \frac{6!}{α! β! γ!} 1^{α} {(2 x)}^{β} \cdot {(\frac{3}{x^{2}})}^{γ}$

$β - 2 γ = 4,$ $α + β + γ = 6 \Rightarrow α = 2 - 3 γ$

$β = 2 γ + 4$ $∴$ $(α, β, γ) = (2, 4, 0)$

$The desired coefficient is$

$\frac{6!}{2! 4!} \cdot 2^{4} = 15 \times 16 = 240$

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