Question

If ${}^n{P}_r{=}^n{P}_{r+1}$ and ${}^n{C}_r{=}^n{C}_{r-1}$, then the values of $n$ and $r$ are

• A. $r,3$
• B. $3,2$
• C. $4,2$
• D. $3,4$

Answer 1

The correct option is A $r,3$

${{}^{n}P}_r={{}^{n}P}_{r+1}\Rightarrow \frac{A!}{r!}=\frac{n!}{(r+1)!}\Rightarrow r+1=0$

${{}^{n}C}_r={{}^{n}C}_{r-1}\Rightarrow \frac{n!(n-r)!}{r!r}=\frac{n!(n-r-1)!}{(r-1)!}$

$(n-r)=r$

$n=2r$

only satisfy 3rd option