If the coefficient of a^7 b^8 in the expansion

$\frac{10 !}{\alpha ! \beta ! \gamma !} \mathrm{a}^{\alpha}(2 \mathrm{~b})^{\beta} \cdot(4 \mathrm{ab})^{\gamma}$

$\frac{10 !}{\alpha ! \beta ! \gamma !} \mathrm{a}^{\alpha+\gamma} \cdot \mathrm{b}^{\beta+\gamma} \cdot 2^{\beta} \cdot 4^{\gamma}$

$\alpha+\beta+\gamma=10$ ............(1)

$\alpha+\gamma=7$    .............(2)

$\beta+\gamma=8$        ...................(3)

$(2)+(3)-(1) \Rightarrow \gamma=5$

$\alpha=2$

$\beta=3$

so coefficients $=\frac{10 !}{2 ! 3 ! 5 !} 2^{3} \cdot 2^{10}$

$=\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5}{2 \times 3 \times 2 \times 5 !} \times 2^{13}$

$=315 \times 2^{16} \Rightarrow \mathrm{k}=315$