Question

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

• A. n = 3, r = 3
• B. n = 3, r = 2
• C. n = 2, r = 3
• D. n = 2, r = 2

Answer 1

Answer: (B)

We have, ${}^n{P}_r={}^n{P}_{r+1}$
$\Rightarrow \frac{n!}{\left(n-r\right)!}=\frac{n!}{\left(n-r-1\right)!}\Rightarrow \frac{1}{\left(n-r\right)}=1$
or $n-r=1$ ....(1)
Also, ${}^n{C}_r{=}^n{C}_{r-1}\Rightarrow r+r-1=n$
$\Rightarrow 2r-n=\mathrm{1.....}(2)$
Solving (1) and (2), we get r = 2 and n = 3