Formula for Combinations With Replacement
I understand how combinations and permutations work (without replacement).
I also see why a permutation of $n$ elements ordered $k$ at a time (with
replacement) is equal to $n^{k}$. Through some browsing I've found that
the number of combinations with replacement of $n$ items taken $k$ at a
time can be expressed as $((\frac{n}{k}))$ [please disregard the division
sign, I don't know how to use the binomial coefficient in MathJax: this
"double" set of parentheses is the notation developed by Richard Stanley
to convey the idea of combinations with replacement].
Alternatively, $((\frac{n}{k}))$ = $(\frac{n+k-1}{k})$ [again without the
division sign: it's a binomial coefficient]. This is more familiar
notation. Unfortunately, I have not found a clear explanation as to why
the above formula applies to the combinations with replacement. Could
anyone be so kind to explain how this formula was developed?
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