Mathematics - Binomial Theorem Question with Solution | TestHub

By the multinomial theorem, the general term is $\frac{10!}{p!q!r!}{2}^q{3}^r{x}^{q+2r}$, where $p+q+r=10$. Coefficient of $x^k={\sum }_{q+2r=k}\frac{10!}{p!q!r!}{2}^q{3}^r$ $a_1(q=1,r=0):10\cdot 2=20\Rightarrow (\mathrm{C})$ is true $a_2:(q,r)=(2,0):45\cdot 4=180;(0,1):10\cdot 3=30$; total $=210\Rightarrow (\mathrm{B})$ is true $a_4:(4,0):210\cdot 16=3360;(2,1):360\cdot 4\cdot 3=4320;(0,2):45\cdot 9=405;$ total $=8085\Rightarrow (\mathrm{D})$ is true $a_6:(6,0):210\cdot 64=13440;(4,1):1260\cdot 16\cdot 3=60480;(2,2):1260\cdot 4\cdot 9=45360;(0,3):120\cdot 27=3240$; total $=122520\ne 12\Rightarrow (\mathrm{A})$ is false