Choose $n$ objects with replacement from a bag with $n$ objects What is the number of ways of choosing $n$ objects with replacement from a bag with $n$ objects (where order doesn't matter and each object is distinct)? The answer is $\binom{2n-1}{n}$ but I don't see why (this is the number of ways to exhaustively enumerate every possible resample of a data set with $n$ observations <

I think that stars and bars) can be used here. For any pair of natural numbers $n$ and $k$, the number of distinct $k$-tuples of non-negative integers whose sum is $n$ is given by the binomial coefficient $$ {n+k-1\choose n}. $$ In this case, $k=n$ and we obtain the answer ${2n-1\choose n}$.