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

Question

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

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

Answer 1

Major Hint:

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?

$~$

https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)