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

Let X = (¹⁰C₁)² + 2(¹⁰C₂)² + 3(¹⁰C₃)² + ·s + 10(¹⁰C₁₀)², where ¹⁰Cᵣ, r ∈ {1, 2, ·s, 10} denote binomial coefficients. Then, the value of 1/1430 X is ______.

$X = \sum_{r=1}^{10} r\,({}^{10}C_r)^{2}$

$\text{To find: } \dfrac{1}{1430}X$

$1.\ r\,{}^{10}C_r = 10\,{}^{9}C_{r-1}$

$2.\ \text{apply Vandermonde's identity}$

$3.\ \text{evaluate then divide by } 1430$

$r\,{}^{10}C_r = 10\,{}^{9}C_{r-1}$

$X = \sum_{r=1}^{10}\left(10\,{}^{9}C_{r-1}\right){}^{10}C_r$

$= 10\sum_{r=1}^{10}{}^{9}C_{r-1}\,{}^{10}C_r$

$s = r-1:\quad X = 10\sum_{s=0}^{9}{}^{9}C_s\,{}^{10}C_{s+1}$

${}^{10}C_{s+1} = {}^{10}C_{9-s}$

$X = 10\sum_{s=0}^{9}{}^{9}C_s\,{}^{10}C_{9-s}$

$= 10\cdot{}^{19}C_9$

${}^{19}C_9 = \dfrac{19!}{9!\,10!} = 92378$

$X = 10 \times 92378 = 923780$

$\dfrac{X}{1430} = \dfrac{10 \times 92378}{1430}$

$1430 = 10 \times 143 \ \Rightarrow\ \dfrac{92378}{143}$

$= 646$

$\text{check: } X = \dfrac{10}{2}\,{}^{20}C_{10} = 5 \times 184756 = 923780$

$\boxed{\dfrac{1}{1430}X = 646}$