A 10-digit number is called *beautiful* if the digits 0, 1, 2, 3, 4 appear in ascending order, and the digits 9, 8, 7, 6, 5 appear in descending order.

If 0 can't be the first digit, how many *beautiful* numbers exist?

**Note:** As an explicit example, 9018762354 is *beautiful* because the 2 sequences appear in order, as evidenced by \( \color{red}{9} 0 1 \color{red}{8 76 } 23 \color{red} {5} 4 \).

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