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

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

Professor PC Thomas Classes · Answer

(315) : We have, $(a+2 b+4 a b)^{10}$\ $=a^{10} \cdot b^{10}\left(\frac{1}{b}+\frac{2}{a}+4ight)^{10}$\ $	herefore T_{r+1}=a^{10} \cdot b^{10} \cdot \frac{10!\cdot\left(\frac{1}{b}ight)^{r_{1}} \cdot\left(\frac{2}{a}ight)^{r_{2}} \cdot(4)^{10-r_{1}-r_{2}}}{r_{1}!r_{2}!\left(10-r_{1}-r_{2}ight)!}$ For coefficient of $a^{7} b^{8}$, put $r_{1}=2, r_{2}=3$\ $	herefore \quad$ Coefficient of $a^{7} b^{8}=\frac{10!(2)^{3} \cdot(4)^{5}}{2!\cdot 3!\cdot 5!}$\ $=2520 	imes 2^{3} 	imes 2^{10}$\ $\Rightarrow \quad K \cdot 2^{16}=315 	imes 2^{3} 	imes 2^{13} \Rightarrow K=315$\