Beautiful numbers

Probability Level 3

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 9018762354 \color{#D61F06}{9} 0 1 \color{#D61F06}{8 76 } 23 \color{#D61F06} {5} 4 .

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