Factorial : n!/(n-k)! = n(n-1)(n-2)...(n-k+1) - why?

  1. Why is the equation

    (A) n!/(n-k)! = n(n-1)(n-2)...(n-k+1)

    true?

    For example, let n=4 and k=2, then

    4!/2! = 4x3x2x1 / 2x1 = 4x3 = 12.

    I understand this example, but I can't make the connection with this and the right-hand-side of equation (A).

    For example, why is our example above not

    4!/2! = 4(4-1)(4-2)...(4-2+1).

    I know this doesn't make any mathematical sense, but I can't understand how the equation on the right-hand-side of (A) is derived.

    Thanks for your help.

    M
     
  2. jcsd
  3. Dick

    Dick 25,735
    Science Advisor
    Homework Helper

    The equation is an informal shorthand. You aren't supposed to include (n-2) as a factor in the case where n=4 and k=2. You are supposed to STOP at (n-k+1)=3.
     
  4. Borek

    Staff: Mentor

    Adding to what Dick wrote - it may become more obvious when you try to derive the equation.

    [tex]\frac {n!} {(n-k)!} = \frac {n \times (n-1) \times (n-2) \times ... \times (n - k + 1) \times (n - k) \times (n - k -1) \times ... \times 3 \times 2 \times 1} {(n-k) \times (n-k-1) \times (n-k-2) \times ... \times 3 \times 2 \times 1} [/tex]

    Check what cancels out and what is left. And remember that when n and k are too small it is not possible to explicitly list all these terms.
     
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