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JEE Main · 2025 · 04 Apr 2025 · Shift 2 · Binomial Theorem

If

1^{2} \cdot (^{15} C_{1}) + 2^{2} \cdot (^{15} C_{2}) + 3^{2} \cdot (^{15} C_{3}) + \dots . + 1 5^{2} \cdot (^{15} C_{15}) = 2^{m} \cdot 3^{n} \cdot 5^{k}

where

m, n, k \in N

, then

m + n + k

is equal to :-

(A)19✓ Correct
(B)21
(C)18
(D)20

Answer: (A)

Solution

Using

r (r 15) = 15 (r - 1 14)

and then

r = (r - 1) + 1

, the sum becomes

15 (14 \cdot 2^{13} + 2^{14}) = 15 \cdot 2^{13} \cdot 16 = 15 \cdot 2^{17} = 2^{17} \cdot 3 \cdot 5

. Hence

m = 17, n = 1, k = 1

and

m + n + k = 19

.

JEE Main · 2025 · 04 Apr 2025 · Shift 2 · Single correct option

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