# Combinatorics proof for identity $\binom{n+t-1}{t-1} = \sum\limits_{k=1}^t \binom{t}{k}\binom{n-1}{k-1}$

$$\binom{n+t-1}{t-1} = \sum\limits_{k=1}^t \binom{t}{k}\binom{n-1}{k-1}$$

how do you prove this? I cant get creative enough to get a combinatorial proof. I have tried picking the problem apart but I can't seem to get a proof at the end

• You need to do a better job typesetting this. For starters, what is the indexing variable on the summation? $n$? $t$? $k$? What does it range between? See this page for formatting information and how to use $\LaTeX$ and MathJax on this site. Sep 15, 2020 at 18:53
• i'm not sure how to use latex but the indexing is k = 1 till t
– mei
Sep 15, 2020 at 18:55
• As for a hint, a binomial coefficient of the form $\binom{a+b-1}{b-1}$ should make you immediately think of Stars and Bars. Meanwhile a summation like this should make you think of breaking into cases based on the size of the indexing variable. Sep 15, 2020 at 18:55
• @JMoravitz i'm kind of new to combinatronics, so i'm not what you mean
– mei
Sep 15, 2020 at 19:00
• Please do not deface your question after it has been answered. Sep 19, 2020 at 21:03

How many ways can you distribute $$n$$ identical balls into $$t$$ distinct baskets where you allow some of the baskets to be empty?
How many ways can you distribute $$n$$ identical balls into $$t$$ distinct baskets where exactly $$k$$ of the baskets are non-empty?
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