QJEEM21Ma0dab: JEE Main 2021 Mathematics - Class 11 JEEM21Ma0dab Solution: If and , then the value of r is equal to :

Question

Verified Solution • 2025–26 Session moderate

If

{}^n{P_r} = {}^n{P_{r + 1}}

and

{}^n{C_r} = {}^n{C_{r - 1}}

, then the value of r is equal to :

Accepted Answer

{}^n{P_r} = {}^n{P_{r + 1}} \Rightarrow {{n!} \over {(n - r)!}} = {{n!} \over {(n - r - 1)!}}

\Rightarrow (n - r) = 1

.....(1)

{}^n{C_r} = {}^n{C_{r - 1}}

\Rightarrow {{n!} \over {r!(n - r)!}} = {{n!} \over {(r - 1)!(n - r + 1)!}}

\Rightarrow {1 \over {r(n - r)!}} = {1 \over {(n - r + 1)(n - r)!}}

\Rightarrow n - r + 1 = r

\Rightarrow n + 1 = 2r

..... (2)

From (1) and (2),

2r - 1 - r = 1 \Rightarrow r = 2

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