What is the number of ways to pick $k$ non-negative integers that add up to $n$? Assume that these are ordered sets.

Question

So the specific question I was asked was to find the number of ways $3$ numbers between $0$ to $99$ can add up to $100$, but I'm curious if this can also be generalized.

So far I've figured out that if we need to add $2$ integers to a number $n$, we simply have $n-1$ different ways as follows:

$1 + n-1$

$2 + n-2$

$3 + n-3$

...

$n-1 + 1$

Edit: I think the stars and bars problem is pretty much the same as my question except im not allowed to use the number 'n' itself. I.e 0+0+100 is not allowed(though 0+50+50 is)

I think my answer would be C(n+k-1, k-1) - k

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, Feb 04 '24 at 07:56
This type of question has been asked & answered on this site many times. Please do a search for earlier appearances. — Gerry Myerson, Feb 04 '24 at 10:09
To be clear, do you consider $20+30+50$ to be different than or the same as $50+20+30$? Do you allow numbers to be the same, i.e. do you count $30+30+40$? See stars and bars or restricted partitions or compositions depending on what restrictions you have. The most likely interpretation is probably stars-and-bars. — JMoravitz, Feb 04 '24 at 16:01
Be aware of the difference between the problems where you allow zero vs where you do not allow zero. — JMoravitz, Feb 04 '24 at 16:02
@JMoravitz my problem allows zeroes but does not allow n itself (0+50+50 is okay but 0+0+100 is not). I think my answer required would be the value from the stars and bars problem, minus 2. Thanks! — iCarry, Feb 04 '24 at 16:16
Minus 3 then since you have 100+0+0, or 0+100+0 or 0+0+100, and minus $k$ in general then. — JMoravitz, Feb 04 '24 at 16:21

What is the number of ways to pick $k$ non-negative integers that add up to $n$? Assume that these are ordered sets.

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