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} $
$m=17 \quad n =1 \quad k =1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "SECTION - A [MATHS - MCQ]" from JEE Main 4-April-2025 Paper - Shift 2→JEE Main 4-April-2025 Paper - Shift 2 — all question types→JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $f(x ) = x^3 - 2x + 2$. If real numbers $a$, $b$ and $c$ such that $\left| {f\left( a \right)} \right| + \left| {f\left( b \right)} \right| + \left| {f\left( c \right)} \right| = 0$ then the value of ${f^2}\left( {{a^2} + \frac{2}{a}} \right) + {f^2}\left( {{b^2} + \frac{2}{b}} \right) - {f^2}\left( {{c^2} + \frac{2}{c}} \right)$ equal to

If $\tan B = \frac{{n\sin A\cos A}}{{1 - n{{\cos }^2}A}}$ then $\tan(A + B)$ equals

If $\frac{{dy}}{{dx}} = \frac{{xy + y}}{{xy + x}}$, then the solution of the differential equation is

The locus of the point of intersection of the perpendicular tangents to the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$ is

If $0\, \le \,x\, < \frac{\pi }{2},$ then the number of values of $x$ for which $sin\,x -sin\,2x + sin\,3x=0,$ is

$\int_{}^{} {\frac{{dx}}{{\sqrt {2x - {x^2}} }} = } $

The position of the point $(8,-9)$ with respect to the lines $2x + 3y - 4 = 0$ and $6x + 9y + 8 = 0$ is

If $2(y - a)$ is the $H.M.$ between $y - x$ and $y - z$, then $x - a,\;y - a,\;z - a$ are in

If $\lambda $ be the ratio of the roots of the quadratic equation in $x, 3m^2x^2 + m(m -4)x + 2 = 0$, then the least value of $m$ for which $\lambda + \frac{1}{\lambda } = 1$, is