If P 1 1 + 2 · P 2 2 + 3 · P 3 3 + … + 15 · P 15 15 = P r q - s ,   0 ≤ s…

We have,

${}^{1}P_1+2·{}^{2}P_2+3·{}^{3}P_3+\dots +15·{}^{15}P_{15}={}^{q}P_r-s$

where, $0\le s\le 1$

Now,

${}^1{P}_1+2{·}^2{P}_2+3{\cdot }^3{P}_3+\dots +15{·}^{15}{P}_{15}$

$=1!+2·2!+3·3!+\dots \dots +15·15!$

$=(2!-1!)+(3!-2!)+(4!-3!)+\dots \dots +(16!-15!)$

$\left[\because n·n!=\left(n+1-1\right)n!=\left(n+1\right)!-n!\right]$

$=16!-1!={}^{q}P_r-\mathrm{s}$

$\Rightarrow q=r=16 \& s=1$

So,

${}^{q+s}C_{r-s}={}^{17}C_{15}=\frac{17\times 16}{2}=136$