1. ## formulas

What is the difference between c(n+r-1,r) and c(n+r-1,r-1)?
When do we use them?

2. ## Re: formulas

Originally Posted by kastamonu
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.

3. ## Re: formulas

Originally Posted by Plato
As far as I know $\dbinom{n+r-1}{r-1}$ has no particular use beyond what it is.
This is the number of ways to put $n$ identical objects into $r$ distinct cells.

Proof:

$\dbinom{n+r-1}{r-1} = \dbinom{n+r-1}{(n+r-1)-(r-1)} = \dbinom{n+r-1}{n} = \dbinom{r+n-1}{n}$

4. ## Re: formulas

Many Thanks. I couldn't find an explanation in my book(CHEN CHUAN-CHONG and KOH KHEE-MENG).

5. ## Re: formulas

Originally Posted by SlipEternal
This is the number of ways to put $n$ identical objects into $r$ distinct cells.
Proof:
$\dbinom{n+r-1}{r-1} = \dbinom{n+r-1}{(n+r-1)-(r-1)} = \dbinom{n+r-1}{n} = \dbinom{r+n-1}{n}$
Did you notice that I was using $n$ distinct cells and $r$ identical objects? You swapped my symbols did you not?

6. ## Re: formulas

Originally Posted by Plato
Did you notice that I was using $n$ distinct cells and $r$ identical objects? You swapped my symbols did you not?
I did notice that. That is why I clarified the swap of symbols and showed how it can be proven that is what something that the formula represents.

7. ## Re: formulas

Do you know/or can recommend a very good combinatorics book that can make the reader a master of the subject?

8. ## Re: formulas

Originally Posted by kastamonu
Do you know/or can recommend a very good combinatorics book that can make the reader a master of the subject?
Almost any Discrete Mathematics book will have the topics you need in it.
I highly recommendDiscrete Mathematics with Combinatorics by James Anderson.

Many Thanks.