Let X = ( ^ 10 C _1 )^2+2 ( ^ 10 C _2 )^2+3 ( ^ 10 C _3 )^2+ . .+… - Vidyadip

MCQ

Let $X =\left({ }^{10} C _1\right)^2+2\left({ }^{10} C _2\right)^2+3\left({ }^{10} C _3\right)^2+\ldots \ldots . .+10\left({ }^{10} C _{10}\right)^2$ where ${ }^{10} C _{ r }, r \in\{1,2, \ldots ., 10\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430} X$ is. . . . . . .

A
$430$
B
$435$
C
$540$
✓
$646$

✓

Answer

Correct option: D.

$646$

d
$X=\sum_{r=0}^{10} r\left({ }^{10} C_r\right)^2$

$X=\sum_{r=0}^{10}(10-r)\left({ }^{10} C_{10-r}\right)^2$

$2 X=\sum_{r=0}^{10} 10\left({ }^{10} C_r\right)^2$

$X=5 \cdot{ }^{20} C_{10} \Rightarrow \frac{X}{1430}=646.00$

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