Power of Combinatorics

A boy is asked to find the number of all possible distinct \(10\)-digit (in Base-\(10\) Notation) positive integers such that in each number, each of the digits \(0,1,2,3,4,5,6,7,8\) and \(9\) is used exactly once. Of course, the leftmost digit in each of these numbers is non-zero.

But he doesn't know Combinatorics. So, he decided to list all such numbers first.

Assume that he can write any single digit in exactly \(1\) second. And neglect the time between completion of writing one digit and starting to write the next one.

How many days will he need to complete the list where each such number appears exactly once?

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