How to enumerate all the places where there are identical ways of choosing concatenated numbers?
$X, Y$ and $Z$ are friends. $X$ comes up with an integer between 1 and $10, Y$ comes up with an integer between 1 and 50 , and $Z$ comes up with an integer between 1 and 100 . The three numbers are then concatenated (combined) to form a larger number $t$ in the order of the numbers chosen by first $X$, then $Y$, and finally $Z$. For example, if $X$ chose $4, Y$ chose 39 , and $Z$ chose $67, t$ would be 43,967 .
Naively, you would think there are 50,000 choices but there are actually 46320 because of the example below.
How do you find all the 3680 identical concatenated numbers?
For example, There are two ways of obtaining the number 7345, Either the first person chooses 7 and the second person chooses three and the last chooses 45 or the first chooses 7, the second chooses 34, and the last person chooses five.