Montag, 3. November 2014

Permutations

  1. Permutations with repetition 
    2.  Choosing from n things offers n choices each time!
    While choosing r of them, the permutations are:

    n · n · n ... (r times)

    Explanation:
    There are n possibilities for the first choice, THEN there are n possibilities for the second choice, and so on, multplying each time.

    We then use the exponent of r to write it down as follow:

    n · n · n ... (r times) = nr (n exponent of r)


    The Formula

    nr

    where:
    1. n is the number of things to choose from;
      2. and
    2. r the chosen things out of n
      3. (Here Repetition is allowed and order matters)

    Example:
    3 numbers are chosen from 10 available (numbered 0, 1, 2, ..., 9)! What could be the permutations?

    The answer:

    10 · 10 · ... (3 times) = 103 = 1.000 permutations



  2. Permutations without Repetition
    3. In this case, the number of available choices is reduced each time we pick a number.

    Example: Assuming we want to know what order 15 pool balls could be in!

    There're numbered
    1, 2, ...., 15: see Img ➦

    It goes like this: if we choose one number, let's say 5, we won't be able to choose aigain! Our next possibily of choosing will be amoung 14 poolballs:
    So, our first choice has 15 possibilities, the next choice 14 possibilities, then comes 13, 12, etc.
    And the total permutations will be:
    15 · 14 · 13 · ... = 15! = 1.307.674.368.000
    Now, we just want to choose 3 of them.
    So the permutation will be:
    15 · 14 · 13 = 2.730
    (Which means, there are 2.730 different ways to arrange 3 pool balls out of 15 balls)

Without repetition the choices get reduced each time.

To better express this mathematically, we use the Factorial function: symbolized !, which means multipying a series of descending natural numbers.
Examples:

  • 5! = 5 · 4 · 3 · 2 · 1 = 120
  • 4! = 4 · 3 · 2 · 1 = 24
  • 3! = 3 · 2 · 1 = 6
  • 2! = 2 · 1 = 2
  • 1! = 1

    • Note: 0! = 1 is an Axiom

To recap:
  1. If we want to select all of the 15 pool balls, the permutations will be:

    2. 15! = 1.307.674.368.000

    and
  2. If we want to select just 3, the permutations will be:

    3. 15!
    (15 − 3)!
    		  = 
    15!
    12!
    		  = 
    15·14·13·12!
    12!
    		  =  15·14·13  =  2.730

Formula and Notations

P(n, r)  =  nPr  =  nPr  = 
n!
(n − r)!
With:
  1. n is the number of things to choose from;

    2. and

  2. r the chosen things out of n
    3. (No repetition, order matters)






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