Formulas for permutations

There are a number of formulas that are used in solving problems related to permutations. The following are the main formulas related to permutations:

Permutations of 'n' objects taken 'r' at a time when all objects are different:

`P_r^n = (n!)/(n - r)!`

Permutations of 'n' objects taken 'r' at a time when one object is always included in each arrangement:

`P = r*P_{r-1}^{n-1}`

Permutations of 'n' objects taken 'r' at a time when one object is always excluded in each arrangement:

`P = P_r^{n-1}`

Permutations of 'n' objects when 'p' are same and of one kind, 'q' are same and of another kind, 'r' are same and of another kind, and the rest 'x' objects are all different:

`P = (n!)/(p! q! r!)`

Circular permutations of 'n' objects:

`P = (n - 1)!`

Circular permutations of 'n' objects when any given object can not have the same two neighboring objects (as in the case of a necklace):

`P = {(n - 1)!}/(2!)`