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

Engineering

Mathematics

Properties of Binomial Coefficients

Question

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

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X = 0 · (¹⁰C₀)² + 1 · (¹⁰C₁)² + 2 · (¹⁰C₂)² + ….… + 10 · (¹⁰C₁₀)²

X = 0 · (¹⁰C₀)² + 9 · (¹⁰C₁)² + 2 · (¹⁰C₂)² + ….… + 0 · (¹⁰C₁₀)²

——————————————————————————

2X = 10 [(¹⁰C₀)² + (¹⁰C₁)² + ….… + (¹⁰C₁₀)²]

2X = 10 · ²⁰C₁₀

$X = 5 \cdot \frac{20!}{10! \cdot 10!} = 5 \frac{11 \cdot 12 \cdot 13 \cdot 14 \cdot 15 \cdot 16 \cdot 17 \cdot 18 \cdot 19 \cdot 20}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10}$

X = 11 · 13 · 17 · 19 · 20 = 1430 × 17 × 19 × 2

$\Rightarrow \frac{X}{1430} = 646.00$