Generate random numbers until every digit [0, 9] is generated

I want to make an experiment where I create a list of many lists of randomly generated sequences that all contain every digit 0 to 9 inclusive, that is, the generation function is to generate random numbers and place them in a list of integers while there is at least 1 digit not found in the list.

The intention for the experiment is to try to make some generalizations about things like expected number # of digits in such a function, how long can a sequence get(can my program loop indefinitely and never find that last digit?), and other interesting things(for me).

I am using PERL for the experiment.

The idea seemed simple at first, I sat down, created a list, and figured I can just make a loop that runs an arbitrary amount of times (I decided to choose 100 times), which calls a function generate_sequence(input: none, output: list of numbers that contains at least 1 of every digit) and adds it to the list.

I quickly realized that I struggle cleanly specifying what it means, pragmatically, to generate a list of numbers that contains one of every digit.

My original attempt was to make a list of digits(0..9), and as I generate numbers, I would search the list for that digit if it is in the list, and remove it. This way, it would generate numbers until the list of digits "still needed" is empty. This approach seems unappealing and can involve a lot of redundant tasks such as checking whether the digit generated is in the list of digits needed every single time a number is generated...

Is there a more elegant solution to such a problem? I am really unhappy with the way I am approaching the function.

In general, I need a function F that accepts nothing, and returns a list of randomly generated numbers that contains every digit 1..9, that is, it stops as soon as every digit from 1 to 9 inclusive is generated.

Thanks ahead of time.