Total solutions: \({17 \choose 4}\text{.}\)
Solutions where \(x_1 > 3\text{:}\) \({13 \choose 4}\text{.}\) Give \(x_1\) 4 units first, then distribute the remaining 9 units to the 5 variables.
Solutions where \(x_1 > 3\) and \(x_2 > 3\text{:}\) \({9 \choose 4}\text{.}\) After you give 4 units to \(x_1\) and another 4 to \(x_2\text{,}\) you only have 5 units left to distribute.
Solutions where \(x_1 > 3\text{,}\) \(x_2 > 3\) and \(x_3 > 3\text{:}\) \({5 \choose 4}\text{.}\)
Solutions where \(x_1 > 3\text{,}\) \(x_2 > 3\text{,}\) \(x_3 > 3\text{,}\) and \(x_4 > 3\text{:}\) 0.