What is the difference between c(n+r-1,r) and c(n+r-1,r-1)?
When do we use them?
The first one is $\dbinom{n+r-1}{r}$ is the number of ways to put $r$ identical objects into $N$ distinct cells.
uses:
- Solve $x_1+c_2+x_3+x_4+x_5=100$ where each $x_k$ is a nonnegative integer. $N=5~\&~r=100$.
- How many ways can six people occupy three rooms in an office? $N=3~\&~r=6$.
- How many ways can banana splits be made from 28 flavors using three scoops each. $N=28~\&~r=3$.
As far as I know $\dbinom{n+r-1}{r-1}$ has no particular use beyond what it is.
Now if we change one word in:
Solve $x_1+c_2+x_3+x_4+x_5=100$ where each $x_k$ is a positive integer.
NOW $N=5~\&~r=95$. Because I think that we start off with one ball in each cell.