Question

If ${}^{12}{P}_5+5\cdot {}^{12}{P}_4={}^{13}{P}_r$, find r.

Answer 1

We have
${}^{(n-1)}{P}_r+r\cdot {}^{(n-1)}{P}_{(r-1)}={}^n{P}_r$ and $r\le n$
${}^{12}{P}_5+5\cdot {}^{12}{P}_4={}^{13}{P}_5={}^{13}{P}_r$(given)
$\Rightarrow r=5$.