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  • If the coefficient of in the expansion of <img alt="(a+2 b+4 a b)^{10} \text { is } K \cdot 2^{16}" src="https://entrancecorner.codecogs.com/gif.latex?

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

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best_answer

General Term  = \frac{10!}{p!q!r!}(a)^{p}(2p)^{q}(4ab)^{r}\\

                     =\frac{10!}{p!q!r!}2^{q+2r}a^{p+r}b^{q+r}\\

where\; p+q+r=10\\   .......(1)

p+r=7\\                          .......(2)

and\; q+r=8\\                ........(3)

(2)+(3)-(1)\Rightarrow r=5, So\; p=2,q=3\\

Coefficient=\frac{10!}{5!3!2!}\times2^{3+2\times5}\\

                          =\frac{10\times9\times8\times7\times6}{3\times2\times2}\times2^{13}=315\times2^{16}\\

                 \Rightarrow k=315

Posted by

Kuldeep Maurya

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