Question Number 191177 by naka3546 last updated on 19/Apr/23
$$\mathrm{Find}\:\:\mathrm{the}\:\:\mathrm{coefficient}\:\:\mathrm{of}\:\:{x}^{\mathrm{18}\:} \:\:\mathrm{in}\:\:\mathrm{expression}\:\: \\ $$$$\left(\mathrm{1}+\:{x}^{\mathrm{3}} \:+\:{x}^{\mathrm{5}} \:+\:{x}^{\mathrm{7}} \right)^{\mathrm{100}} \:\:. \\ $$
Answered by mehdee42 last updated on 19/Apr/23
$$\left(_{\mathrm{94},\mathrm{6}} ^{\mathrm{100}} \right)+\left(_{\mathrm{96},\mathrm{2},\mathrm{1},\mathrm{1}} ^{\:\:\:\:\:\mathrm{100}} \right)+\left(_{\mathrm{96},\mathrm{1},\mathrm{3}} ^{\:\:\mathrm{100}} \right)=\frac{\mathrm{100}!}{\mathrm{94}!\mathrm{6}!}+\frac{\mathrm{100}!}{\mathrm{96}!\mathrm{2}!}+\frac{\mathrm{100}!}{\mathrm{96}!\mathrm{3}!} \\ $$
Commented by naka3546 last updated on 19/Apr/23
$$\mathrm{Thank}\:\mathrm{you} \\ $$