If , where m, n, k N, then m + n + k is equal to : [2025]

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Solution

Q.
If $1^{2} \cdot ({}^{15}C_{1}) + 2^{2} \cdot ({}^{15}C_{2}) + 3^{2} \cdot ({}^{15}C_{3}) + . . . + 15^{2} \cdot ({}^{15}C_{15}) = 2^{m} \cdot 3^{n} \cdot 5^{k}$ , where m, n, k $\in$ N, then m + n + k is equal to : [2025]

1 21

2 18

3 20

4 19

Ans.
(4)

Using formula, $\sum_{r = 1}^{n} r^{2} {}^{n}C_{r} = n (n + 1) 2^{n - 2}$

For n = 15

$\sum_{r = 1}^{15} r^{2} {}^{15}C_{r} = 15 \times 16 \times 2^{13}$

= $3 \times 5 \times 2^{4} \times 2^{13}$

= $2^{17} \times 3^{1} \times 5^{1}$

On comparing terms, we get m = 17, n = 1 and k = 1

Thus, m + n + k = 19.

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