Permutations Counting where order matters If you have two tasks T 1 and T 2 that are performed in sequence. T 1 can be performed in n ways. T 2 can be performed in n ways. The sequence T 1 T 2 can be performed in (n 1 )(n 2 ) ways.

3 possible ways 4 possible ways to to perform T 1 perform T 2 Possible ways of performing Task 1 then Task = 12 This is called the multiplication principle of counting.

Lets say you have a password that is one letter followed by 3 digits. How many possible unique passwords can you create? 26 letters in the alphabet 10 possible numbers (0-9) for each digit letters x digit 1 x digit 2 x digit 3 26 x 10 x 10 x 10 = 26000

nPr symbolizes the number of permutations of n objects taken r at a time. If 1 ≤ r ≤ n then: nPr is: n x (n-1) x (n-2)…..(n-r+1) 1 at a time 2 at a time 3 at a time

The number of permutations of 4 objects taken 3 at a time = 4 x 3 x 2 = 24 (n) x (n-1) x (n-2) 4 x 3 x 2 = 24 Taken at a time: Taken 3 at time.

The sequences 12, 43, 31, 24 and 21 are some permutations of set A taken 2 at a time. This is noted as 4P2 The number of permutations of 4 objects taken 2 at a time = 4 x 3 = 12 (n) x (n-1) taken at a time 1 2

To further define nPr, we have a formula nPr = n!/(n-r)! ! Means factorial 3! = 1 x 2 x 3 = 6

If we had a deck of cards, set A = 52 and 5 cards are dealt. The number of permutations of A taken 5 at a time is: 52P5 = 52!/47! = 311,875,200 Or 52 x 51 x 50 x 49 x 48 (n) x (n-1) x (n-2) x (n-3) x (n-4) Taken at a time:

How many words of three distinct letters can be formed form the letters of the work MAST? 4P3 = 4!/(4-3)! = 4!/1! = 24

How many distinguishable words can be formed with the letters of MISSISSIPPI? There are 11 letters 1 M, 4 I, 4 S, 2 P 11!/(1!) (4!) (4!) (2!) = /(1)(24)(24)(2) = /1152 = 34650