How are draws without replacement formalized? Suppose I have some i.i.d. random variables X_1,…,X_n which represent the nth balls drawn uniformly from a bin without replacement.

How are draws without replacement formalized?
Suppose I have some i.i.d. random variables ${X}_{1},\dots ,{X}_{n}$ which represent the nth balls drawn uniformly from a bin without replacement. Is sampling without replacement a condition that is put on the set I define my probability measure on? In that all elements $w\in \mathrm{\Omega }$ are such that they do not have repeated draws? Or is it a property of the random variable with which I use to sample?
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Waldruhylm
Step 1
It will be defined by the sample space, since you are only allowing outcomes that have no repeating elements.
Step 2
So your sample space consists of $\left(\genfrac{}{}{0}{}{K}{n}\right)$ unique selections of balls times n! ways of arranging them or a total size of $\left(\genfrac{}{}{0}{}{K}{n}\right)n!$