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Help with basic binomial coefficient

  1. Jan 19, 2017 #1
    < Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >

    Hello. I'm currently working my way through Lang's Basic Mathematics and cannot make sense of this question:

    Show that if n is a positive integer at most equal to m, then


    [tex]{m \choose n}+{m\choose n-1}={m+1 \choose n}[/tex]

    __

    The answer in the textbook is given as:

    1. [tex]{m \choose n}+{m\choose n-1}={m! \over n!(m-n)!}+{m! \over (m-n+1)!(n-1)}[/tex]

    [common denominator n!(m — n + 1)!]

    2. [tex]= {m!(m-n+1)+m!n\over n!(m-n+1)!}[/tex]
    __

    I omitted the rest of the answer as I understand what follows from 2.

    However I don't understand how to get such denominator from 1.
    Could someone please help me?
     
    Last edited by a moderator: Jan 19, 2017
  2. jcsd
  3. Jan 19, 2017 #2

    mfb

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    2016 Award

    Staff: Mentor

    I moved your thread to our homework section, as it is homework-like.

    In (1), you can write n! as n(n-1)! and (m-n+1)! as (m-n+1)(m-n)!. Afterwards the fractions should be easy to add with the usual methods.
     
  4. Jan 19, 2017 #3
    Thank you for the quick reply. This makes sense!
     
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