Question

The coefficient of $x^5{y}^2{z}^2$ in the expansion of $(x+y+2z{)}^9$, is

• A. 756
• B. 2016
• C. 3024
• D. 6048

Answer 1

Answer: (C)

$T_{r+1}\text{ in }(x+(y+2z){)}^9\text{ is given by }$
$T_{r+1}={}^9{C}_r{x}^{9-r}(y+2z{)}^r$
$\text{ Put }r=4,T_5={}^9{C}_4{x}^5(y+2z{)}^4={}^9{C}_4{x}^5\left[{}^4{C}_p{y}^{4-p}(2z{)}^p\right]$
$\text{ Put }p=2,T_5={}^9{C}_4{}^4{C}_2{x}^5{y}^2{2}^2{z}^2$
Hence coefficient of $x^5{y}^4{z}^2={}^9{C}_4\cdot {}^4{C}_2\cdot 2^2=\frac{9!}{4!5!}\cdot \frac{4!}{2!2!}{2}^2=9\cdot 8\cdot 7\cdot 6=3024$