Question

The coefficient of x⁵ in the expansion of ${\left(1+x\right)}^{21}+{\left(1+x\right)}^{22}+...+{\left(1+x\right)}^{30}$
(a) ⁵¹C₅
(b) ⁹C₅
(c) ³¹C₆ − ²¹C₆
(d) ³⁰C₅ + ²⁰C₅

Answer 1

(c) ³¹C₆ − ²¹C₆
$$\begin{gathered} \mathrm{We}\mathrm{have}{\left(1+x\right)}^{21}+{\left(1+x\right)}^{22}+...{\left(1+x\right)}^{30} \\ ={\left(1+x\right)}^{21}\left[\frac{{\left(1+x\right)}^{10}-1}{\left(1+x\right)-1}\right] \\ =\frac{1}{x}\left[{\left(1+x\right)}^{31}-{\left(1+x\right)}^{21}\right] \\ \mathrm{Coefficient}\mathrm{of}{x}^5\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expansion}=\mathrm{Coefficient}\mathrm{of}{x}^5\mathrm{in}\frac{1}{x}\left[{\left(1+x\right)}^{31}-{\left(1+x\right)}^{21}\right] \\ =\mathrm{Coefficient}\mathrm{of}{x}^6\mathrm{in}\left[{\left(1+x\right)}^{31}-{\left(1+x\right)}^{21}\right] \\ {=}^{31}{C}_6{-}^{21}{C}_6 \end{gathered}$$