If 1^2 (^15C_1)+2^2 (^15C_2)+3^2 (^15C_3)+ +15^2 (^15C_15)=2^m 3^n 5^k, where m,n,k N, the…

Question

If ${1}^{2}\cdot ({}^{15}{C}_{1})+{2}^{2}\cdot ({}^{15}{C}_{2})+{3}^{2}\cdot ({}^{15}{C}_{3})+\ldots +{15}^{2}\cdot ({}^{15}{C}_{15})={2}^{m}\cdot {3}^{n}\cdot {5}^{k}$, where $m,n,k\in N$, then $m+n+k$ is equal to :

Accepted Answer

$\sum _{15}^{r=1}{r}^{2}\cdot {}^{15}{C}_{r}\, ({r}^{n}{C}_{r}={n}^{n-1}{C}_{r-1})$

$\begin{gathered} =15\sum _{r=1}^{15}r\cdot {}^{14}{C}_{r-1} \ =15\sum _{r=1}^{15}(r-1+1){}^{14}{C}_{r-1} \ =15\cdot \sum _{r=1}^{15}(r-1){}^{14}{C}_{r-1}+15\cdot \sum _{r=1}^{15}{}^{14}{C}_{r-1} \ =15\cdot 14\cdot {2}^{13}+15\cdot {2}^{14} \ =15\cdot {2}^{14}(7+1) \ =5\cdot 3\cdot {2}^{17} \ n+m+k=17+1+1=19 \end{gathered}$

Solutions · 1