Results 1 to 5 of 5

Math Help - Prove that a variant of binomical coefficient is integer

  1. #1
    Junior Member
    Joined
    Jun 2010
    Posts
    59

    Prove that a variant of binomical coefficient is integer

    I am familiar with the standard proof that the binomial coefficient is an integer based on induction and Pascal's Rule.

    I'm trying to prove that a related expression is an integer:

    f(m, n) = \frac{(m + n)!}{m!n!}

    Using an adaptation of Pascal's Rule, we can see that f(m,n) + f(m+1,n-1) = f(m+1,n) and f(m - 1,n) + f(m,n - 1) = f(m,n), but these equalities don't lead to an inductive proof.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Prove that a variant of binomical coefficient is integer

    Quote Originally Posted by VinceW View Post
    I'm trying to prove that a related expression is an integer:
    f(m, n) = \frac{(m + n)!}{m!n!}.
    Well \binom{m+n}{n} = \frac{(m + n)!}{m!n!}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2010
    Posts
    715
    Thanks
    2

    Re: Prove that a variant of binomical coefficient is integer

    Forgive me if I'm missing something, but isn't it just

    \frac{(m+n)!}{m! n!} = \frac{(m+n)!}{m! (m+n-m)!} = \binom{m+n}{m} \in\mathbb{N}?

    EDIT: with Plato's post above, I'm definitely not missing something!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Jun 2010
    Posts
    59

    Re: Prove that a variant of binomical coefficient is integer

    Oh, yes, this is mathematically equivalent to the binomial coefficient, but I was hoping to find a more direct proof rather than converting to bionomial coefficient, and falling back on that standard proof.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Prove that a variant of binomical coefficient is integer

    Quote Originally Posted by VinceW View Post
    I am familiar with the standard proof that the binomial coefficient is an integer based on induction and Pascal's Rule.

    I'm trying to prove that a related expression is an integer:

    f(m, n) = \frac{(m + n)!}{m!n!}

    Using an adaptation of Pascal's Rule, we can see that f(m,n) + f(m+1,n-1) = f(m+1,n) and f(m - 1,n) + f(m,n - 1) = f(m,n), but these equalities don't lead to an inductive proof.
    Hint: Try to construct a counting problem which f(m,n) is her answer.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: July 14th 2010, 02:16 PM
  2. let n >= 4 be an integer, prove..
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: April 18th 2009, 09:54 AM
  3. [SOLVED] Prove (a + b) is not an integer
    Posted in the Algebra Forum
    Replies: 2
    Last Post: February 12th 2009, 04:38 AM
  4. Prove this is not an integer
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: February 4th 2008, 07:31 PM
  5. Replies: 2
    Last Post: October 14th 2007, 05:32 AM

Search Tags


/mathhelpforum @mathhelpforum