JEE Main 2025 Apr 4 Shift 2, Mathematics Q13: Properties Of Binomial Coefficient

JEE Main2025Apr 4, Shift 2Mathematics

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
B
21
C
18
D
20

Solution

Use $r \cdot^{n} C_{r} = n \cdot^{n - 1} C_{r - 1}$ . Then

$r = 1 \sum 15 r^{2}^{15} C_{r} = 15 r = 1 \sum 15 r^{14} C_{r - 1} = 15 r = 1 \sum 15 (r - 1 + 1)^{14} C_{r - 1}$

$= 15 [\sum_{r = 2}^{15} (r - 1)^{14} C_{r - 1} + \sum_{r = 1}^{15}^{14} C_{r - 1}]$

$= 15 [14 \cdot 2^{13} + 2^{14}] = 15 \cdot 2^{13} (14 + 2) = 15 \cdot 16 \cdot 2^{13}$

$= 15 \cdot 2^{17} = 3 \cdot 5 \cdot 2^{17}$ .

So $m = 17, n = 1, k = 1$ , giving $m + n + k = 19$ .

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JEE Main 2025 Apr 4 Shift 2, Mathematics Q13: Properties Of Binomial Coefficient

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