Question

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

Answer 1

Answer: 315

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