Pascal's coefficient identity

\[\begin{array}{rc} 0^\text{th} \text{ row:} & 1 \\ 1^\text{st} \text{ row:} & 1 \quad 1 \\ 2^\text{nd} \text{ row:} & 1 \quad 2 \quad 1 \\ 3^\text{rd} \text{ row:} & 1 \quad 3 \quad 3 \quad 1 \\ 4^\text{th} \text{ row:} & 1 \quad 4 \quad 6 \quad 4 \quad 1 \\ \vdots \ \ \ & \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \end{array} \]

Pascal's triangle is shown above for the \(0^\text{th}\) row through the \(4^\text{th}\) row. What is the \(4^\text{th}\) element in the \(10^\text{th}\) row?

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Note: Each row starts with the \(0^\text{th}\) element. For example, the \(0^\text{th}\), \(1^\text{st}\), \(2^\text{nd}\), and \(3^\text{rd}\) elements of the \(3^\text{rd}\) row are 1, 3, 3, and 1, respectively.

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