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)


    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.

  2. jcsd
  3. Dick

    Dick 25,910
    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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