If 1^2 ( ^ 15 C_1 )+2^2 ( ^ 15 C_2 )+3^2 ( ^ 15 C_3 )+ .+15^2 ( ^… - Vidyadip

MCQ

If $1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots .+15^2 \cdot\left({ }^{15} C_{15}\right)=$ $2^{ m } \cdot 3^{ n } \cdot 5^{ k }$,
where $m , n , k \in N$, then $m + n + k$ is equal to :-

✓
19
B
21
C
18
D
20

✓

Answer

Correct option: A.

19

(A) 19
$\sum_{ r =1}^{15} r ^2\left({ }^{15} C _{ r }\right) \Rightarrow 15 \sum_{ r =1}^{15} r ^{14} C _{ r -1} $
$15 \sum_{ r =1}^{15}( r -1+1){ }^{14} C _{ r -1} $
$15 \cdot 14 \sum_{ r =1}^{15}{ }^{13} C _{ r -2}+15 \sum_{ r =1}^{14} C _{ r -1} $
$15 \cdot 14 \cdot 2^{13}+15 \cdot 2^{14} $
$3^1 \cdot 2^{13}(70+10) $
$3^1 \cdot 2^{13} \cdot 80 $
$3^1 \cdot 5^1 \cdot 2^{17} $
$m=17 \quad n =1 \quad k =1$

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