If the coefficient of a⁷b⁸ in the expansion of (a + 2b + 4ab)¹⁰ is K.2¹⁶, then K is equal to _____________.

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