Question

If the coefficient of $a ^{7} b ^{8}$ in the expansion of $(a+2 b+4 a b)^{10}$ is $K \cdot 2^{16}$, then $K$ is equal to _____

Answer 1

$\frac{10 !}{\alpha ! \beta ! \gamma !} a ^{\alpha}(2 b )^{\beta} \cdot(4 ab )^{\gamma}$
$\frac{10 !}{\alpha ! \beta ! \gamma !} a ^{\alpha+\gamma} \cdot b ^{\beta+\gamma} \cdot 2^{\beta} \cdot 4^{\gamma}$
$\alpha+\beta+\gamma=10 ....$(1)
$\alpha+\gamma=7 ....$(2)
$\beta+\gamma=8 ....$(3)
$(2)+(3)-(1) \Rightarrow \gamma=5$
$\alpha=2$
$\beta=3$
so coefficients $=\frac{10 !}{2 ! 3 ! 5 !} 2^{3} \cdot 2^{10}$
$=\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5}{2 \times 3 \times 2 \times 5 !} \times 2^{13}$
$=315 \times 2^{16} \Rightarrow k =315$