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JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Permutations and Combinations

If

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

and

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

, then the value of

r

is equal to:

(A)4
(B)1
(C)3
(D)2✓ Correct

Answer: (D)

Solution

From

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

,

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

, so

(n - r)! = (n - r - 1)!

and hence

n - r = 1

. From

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

with

r \neq = r - 1

, the two lower indices must add to

n

, so

r + (r - 1) = n

. Substituting

n = r + 1

gives

r + 1 = 2 r - 1

, so

r = 2

.

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Single correct option

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