Question

If ${{}^{n}P}_r={{}^{n}P}_{r+1}$ and ${{}^{n}C}_r={{}^{n}C}_{r-1}$, thn find $(n,r)$.

Answer 1

${}^n{P}_r{=}^n{P}_{r+1}$

$\therefore \frac{n!}{(n-r)!}=\frac{n!}{(n-r-1)!}$

$\therefore (n-r)(n-r-1)!=(n-r-1)!$

$\therefore n-r=1$........(1)

$\therefore \frac{n!}{(n-r)!r!}=\frac{n!}{(n-r+1)!(r-1)!}$

$\frac{1}{r}=\frac{1}{(n-r+1)}$

$\therefore n+1=2r$.......(2)

$\therefore$ From (1) & (2)

$1+r+1=2r$

$\therefore 2=r$

$\therefore n=3$

Ans :- $r=2$ & $n=3$