Question

If the sum $S{=}^{20}{C}_2+{2.}^{20}{C}_3+{3.}^{20}{C}_4+.....+{19.}^{20}{C}_{20}$ can be expressed as $1+p.2^q$, then find the minimum value of $(p+q)$ where $p$ and $qϵN$.

Answer 1

$S={\sum }_{r=1}^{20}{(r-1)}^{20}{C}_r$

$={\sum }_{r=1}^{20}{}^{20}{C}_r-{\sum }_{r=1}^{20}{}^{20}{C}_r$

$=20{\sum }_{r=1}^{20}{}^{19}{C}_{r-1}-(2^{20}-1)$

$=20\left(2^{19}\right)-\left(2^{20}-1\right)$

$=1+\left(2^{19}\right)(20-2)$

$=1+18\left(2^{19}\right)$

$\Rightarrow p=18$

$\Rightarrow q=19$

$\Rightarrow p+q=37$.