If ^n P_r=^n P_r+1 and ^n C_r=^n C_r-1 , then the values of n and r are

Correct answer: (B) 3,2. We have, ^n P_r=^n P_r+1 ⇒ (n !)/((n-r) !)=(n !)/((n-r-1) !) ⇒ 1/((n-r))=1 or n-r=1 Also,^n C_r=^n C_r-1 ⇒ r+r-1=n ⇒ 2 r-n=1 Solving (1) and (2), we get r=2 and n=3

Q46 / 110 · VITEEE 2023 · Mathematics

If \({ }^n P_r={ }^n P_{r+1}\) and \({ }^n C_r={ }^n C_{r-1}\), then the values of \(\mathrm{n}\) and \(\mathrm{r}\) are

A
4,3
Correct answer
B
3,2
C
4,2
D
None of these

Solution

We have, \({ }^n P_r={ }^n P_{r+1}\) \(\begin{aligned} & \Rightarrow \frac{n !}{(n-r) !}=\frac{n !}{(n-r-1) !} \Rightarrow \frac{1}{(n-r)}=1 \\ & \text { or } n-r=1 \\ & \text { Also, }{ }^n C_r={ }^n C_{r-1} \Rightarrow r+r-1=n \\ & \Rightarrow 2 r-n=1 \end{aligned}\) Solving (1) and (2), we get \(r=2\) and \(n=3\)

Hence, the correct answer is option: 'B'.

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