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Tone2 Gladiator v2.2 review and bonus updates and activator. Tone2 actually has three great, powerful instruments: Gladiator, Kontakt…Q:

A collection of random numbers where the probability of picking a number is a unique property of that number

Consider a collection of $20$ distinct numbers (say $1,2, \ldots, 20$). We can randomly pick one number from the collection and see that each number is equally likely to be selected. I am interested in what happens if we consider a different model.
Instead of $20$ distinct numbers, consider a collection of $10$ distinct numbers (say $1,2, \ldots, 10$). We can randomly pick one number from the collection and see that each number is equally likely to be selected. I am interested in what happens if we consider a different model.

The number of numbers does not matter (i.e., the collection can be as big as desired).
The ordering of the $10$ numbers does not matter (i.e., the order of the numbers in the collection can be shuffled)
We do not pick $k$ distinct numbers to include in the $10$ numbers (say pick $5$ distinct numbers)

The question is:
Is there a unique number that is selected in this scenario? Is there a deterministic way of picking $1,2, \ldots, 10$ numbers that will guarantee a unique number is selected?
In my sample run, $19, 4, 5, 11, 9, 11, 3, 7, 5, 1, 10, 2, 2$ were picked in that order. $19$ was picked $14$ times, $4$ was picked $11$ times, etc.
On average $19/14$ unique numbers are picked (1.143)
The lowest number picked is $10$ (there are $7$ unique numbers that are below $10$).
A related question is:
What is the expected number of picks in the model where we include the $k$ distinct numbers that must be picked, but the order does not matter. We might have to average over how $k$ is selected (such as $5,4,3,3,2,2,2,1,1,1$ etc.)
It might also be helpful to know where to find more information about this question.
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