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Concept behind this factorial

  1. Apr 1, 2013 #1
    Why is it that (x + y)!=(x + y)(x + y - 1)(x + y - 2)...(x + 1)x!
    Where did the last "x!" come from?

    Thanks
     
  2. jcsd
  3. Apr 1, 2013 #2

    jbunniii

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    $$(x+y)! = (x+y)(x+y-1)(x+y-2)\ldots(x+1)(x)(x-1)\ldots(2)(1)$$
    Now just rewrite the rightmost factors ##(x)(x-1)\ldots(2)(1)## as ##x!##.
     
  4. Apr 1, 2013 #3
    Thank you for your quick reply. I got the form you required, but still I have the concept missing. If you don't mind explaining why did we multiply by (x+1)(x)(x−1)…(2)(1)? It seems like we get to a place where y disappears by subtraction but then again why did we add the term (x+1) and so on?
     
  5. Apr 1, 2013 #4

    jbunniii

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    The definition of the factorial of any number ##n## is ##(n)(n-1)\ldots(2)(1)##, i.e., you must keep subtracting until you get all the way down to ##1##. Therefore, when calculating ##(x+y)!##, you don't stop when you get to ##x##; you must continue all the way to ##1##.
     
  6. Apr 1, 2013 #5

    jbunniii

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    Try it with a concrete example if it's still unclear. For example, if ##x = 3## and ##y = 4##, then ##x+y = 7##, and ##(x+y)! = 7! = (7)(6)(5)(4)(3)(2)(1) = (7)(6)(5)(4)3!##.
     
  7. Apr 1, 2013 #6
    Thank you a lot!
     
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