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Thread: formulas

  1. #1
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    formulas

    What is the difference between c(n+r-1,r) and c(n+r-1,r-1)?
    When do we use them?
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  2. #2
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    Re: formulas

    Quote Originally Posted by kastamonu View Post
    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.
    Last edited by Plato; May 5th 2017 at 09:41 AM.
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  3. #3
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    Re: formulas

    Quote Originally Posted by Plato View Post
    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:

    $\dbinom{n+r-1}{r-1} = \dbinom{n+r-1}{(n+r-1)-(r-1)} = \dbinom{n+r-1}{n} = \dbinom{r+n-1}{n}$
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  4. #4
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    Re: formulas

    Many Thanks. I couldn't find an explanation in my book(CHEN CHUAN-CHONG and KOH KHEE-MENG).
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  5. #5
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    Re: formulas

    Quote Originally Posted by SlipEternal View Post
    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?
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  6. #6
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    Re: formulas

    Quote Originally Posted by Plato View Post
    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.
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  7. #7
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    Re: formulas

    Do you know/or can recommend a very good combinatorics book that can make the reader a master of the subject?
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  8. #8
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    Re: formulas

    Quote Originally Posted by kastamonu View Post
    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.
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  9. #9
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    Re: formulas

    Many Thanks.
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