Question

Let $X = \left(^{10}C_{1}\right)^{2}+2\left(^{10}C_{2}\right)^{2}+3\left(^{10}C_{3}\right)^{2}+\cdots+10\left(^{10}C_{10}\right)^{2}$,
where $^{10}C_{r}, \,r\,\in\,\left\{1, 2, \cdots, 10\right\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}$ X is _____ .

Answer 1

$X=\displaystyle\sum_{r=0}^{n} r \cdot\left({ }^{n} C_{r}\right)^{2} ; n=10$
$X=n \cdot \displaystyle\sum_{r=0}^{n}{ }^{n} C_{r} \cdot{ }^{n-1} C_{r-1}$
$X=n \cdot \displaystyle\sum_{r=0}^{n}{ }^{n} C_{n-r} \cdot{ }^{n-1} C_{r-1}$
$X=n \cdot{ }^{2 n-1} C_{n-1} ; n=10 $
$X=10 .{ }^{19} C_{9}$
$\frac{X}{1430}=\frac{1}{143} \cdot{ }^{19} C_{9}=646$