Permutations
These are the different ways in which a collection of items can be arranged.
The different ways in which the alphabets A, B and C can be grouped together, taken all at a time, are:
- ABC
- ACB
- BCA
- CBA
- CAB
- BAC.
Note that ABC and CBA are not same as the order of arrangement is different. The same rule applies while solving any problem in permutations-- with permutations, order matters.
The number of ways in which n things can be arranged, taken all at a time, nPn = n!, called ‘n factorial.’
Factorial Formula
The factorial of a number n is defined as the product of all the numbers from n to 1. For example, the factorial of 5, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Therefore, the number of ways in which the 3 letters can be arranged, taken all a time, is 3! = 3 x 2 x 1 = 6 ways.
The number of permutations of n things, taken r at a time, is denoted by:
nPr = n! / (n-r)!
For example: The different ways in which the 3 letters, taken 2 at a time, can be arranged is 3!/(3-2)! = 3!/1! = 6 ways.
Important Permutation Formulas
• 1! = 1
• 0! = 1