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Small question about binomial theorem

  1. Jun 9, 2012 #1
    I was trying to make sense of the equation attached below which was on the wikipedia site.

    However I'm not entirely sure how to make use of the "n choose 0" , "n choose 1", etc. statements that in front of each term in of the expansion. I roughly know how the expansion should look intuitively but I was hoping I could find a greater understanding.

    Thank you.
     

    Attached Files:

  2. jcsd
  3. Jun 9, 2012 #2
    Remember that binomial coefficients can be expressed in terms of factorials.

    [tex]\binom{n}{m} = \frac{n!}{m!(n-m)!}[/tex]

    For example,

    [tex]\binom{6}{2} = \frac{6!}{2!4!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(4 \times 3 \times 2 \times 1)} = \frac{6 \times 5}{2} = 15[/tex]

    You see that the 4! on the bottom canceled all but two factors on the top. One of the terms can always be used to get a lot of cancellations in this way.
     
  4. Jun 9, 2012 #3

    Ray Vickson

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    [tex] {n \choose m} \equiv \frac{n!}{m! (n-m)!} = \frac{n(n-1)...(n-m+1)}{m!},[/tex]
    so
    [tex] {n \choose 0} = 1, \: {n \choose 1} = n, \; {n \choose 2} = \frac{n(n-1)}{2}, \; \cdots, {n \choose n} = 1.
    [/tex]
    In practice it is often easier to get them recursively from
    [tex] {n \choose 0} = 1, \: {n \choose 1} = n, \\
    {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}, \; 1 \leq k \leq n.[/tex]

    RGV
     
  5. Jun 10, 2012 #4
    Thank you it is much clearer now.
     
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