$\begin{array} { ccccccc } 1234 & 1243 & 1324 & 1342 & 1423 & 1432 \\ \vdots & & & & & \vdots \\ 4123 & 4132 & 4213 & 4231 & 4312 & 4321 \\ \end{array}$

The 24 distinct 4-digit numbers above have been listed out by arranging 1, 2, 3, and 4 in different orders. What is the sum of these 24 numbers?

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**Hint**: You do not need to list out all 24 numbers to find the sum.