Question Number 48443 by rahul 19 last updated on 24/Nov/18
$${Coefficient}\:{of}\:{x}^{\mathrm{4}} \:{in}\:\left(\mathrm{2}−{x}+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{6}} =? \\ $$
Commented by rahul 19 last updated on 24/Nov/18
$${Do}\:{by}\:{multinomial}\:{expansions}…. \\ $$
Commented by Meritguide1234 last updated on 24/Nov/18
$${rahul}\:{your}\:{post}\:{are}\:{nice}\:{post}\:{more}\:{questions}\:{like}\:{this} \\ $$
Commented by rahul 19 last updated on 24/Nov/18
$${Sure}\:{sir}! \\ $$$${I}'{ll}\:{post}\:{tomorrow}… \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 24/Nov/18
$$\frac{\mathrm{6}!}{{a}!{b}!{c}!}\left(\mathrm{2}\right)^{{a}} \left(−{x}\right)^{{b}} \left(\mathrm{3}{x}^{\mathrm{2}} \right)^{{c}} \\ $$$$=\frac{\mathrm{6}!}{{a}!{b}!{c}!}\left(\mathrm{2}\right)^{{a}} \left(\mathrm{3}\right)^{{c}} \left(−\mathrm{1}\right)^{{b}} \left({x}\right)^{{b}+\mathrm{2}{c}} \\ $$$${a}+{b}+{c}=\mathrm{6} \\ $$$${b}+\mathrm{2}{c}=\mathrm{4} \\ $$$${b}=\mathrm{0}\:\:\:{c}=\mathrm{2}\:\:{so}\:{a}=\mathrm{4} \\ $$$${b}=\mathrm{2}\:\:\:{c}=\mathrm{1}\:\:\:\:{a}=\mathrm{3} \\ $$$${b}=\mathrm{4}\:\:\:\:{c}=\mathrm{0}\:\:\:\:{a}=\mathrm{2} \\ $$$${so}\:{required}\:{coefficient}\:{is} \\ $$$$\left[\frac{\mathrm{6}!}{\mathrm{4}!\mathrm{0}!\mathrm{2}!}×\left(\mathrm{2}\right)^{\mathrm{4}} \left(\mathrm{3}\right)^{\mathrm{2}} \left(−\mathrm{1}\right)^{\mathrm{0}} +\frac{\mathrm{6}!}{\mathrm{3}!\mathrm{2}!\mathrm{1}!}×\left(\mathrm{2}\right)^{\mathrm{3}} \left(\mathrm{3}\right)^{\mathrm{1}} \left(−\mathrm{1}\right)^{\mathrm{2}} +\right. \\ $$$$ \\ $$$$\left.\frac{\mathrm{6}!}{\mathrm{2}!\mathrm{4}!\mathrm{0}!}×\left(\mathrm{2}\right)^{\mathrm{2}} \left(\mathrm{3}\right)^{\mathrm{0}} \left(−\mathrm{1}\right)^{\mathrm{4}} \right] \\ $$$$\:{pls}\:{check}\:{and}\:{calculate}… \\ $$$$\left[\mathrm{15}×\mathrm{16}×\mathrm{9}+\mathrm{60}×\mathrm{8}×\mathrm{3}+\mathrm{15}×\mathrm{4}\right. \\ $$$$=\:\:\:\:\:\:\mathrm{2160}\:\:\:\:\:+\mathrm{1440}+\mathrm{60} \\ $$$$=\mathrm{3660} \\ $$
Commented by rahul 19 last updated on 24/Nov/18
=3660. (2160+1440+60)...I forgot the last case!
thanks sir
Commented by tanmay.chaudhury50@gmail.com last updated on 24/Nov/18
$${thank}\:{you}\:{for}\:{recheck}\:{in}\:{calculation} \\ $$$${i}\:{have}\:{used}\:\mathrm{3}!\:{twice}\:{by}\:{mistake}… \\ $$