A permutation is an arrangement of things with the order being important. abcd is different to bcda.  A combination is a selection of things e.g. abcd is the collection of the four letters with no relevance given to the order.   If a team of eleven players are selected for a football team out of a selection of 20 persons with no importance given to the positions then the selection is combination.  However if each position (goalkeeper, left fullback etc. ) is chosen seperately then the selection is a permutation.

n P(r)  or   n P (r) = Permutation of n items taken r at a time n C(r)  or   n C (r) = Combination of n items taken r at a time

Permutation illustrates the number of ways to arrange elements in a definite order.
The number of permutations of n items taken all at a time =

n! = 1.2.3 .....n     (N Factorial)

The number of permutations of n elements of same kind taken r at a time.

nP(r) = n ! /(n-r)!

Example
The number of permutations of four l etters a,b,c,d arranged three at a time is
4! /(4-3)! = 4.3.2.1 / 1 = 24

a,b,c / a,b,d /a,c,b /a,c,d / a,d,b/ a,d,c
b,a,c / b,a,d /b,c,a /b,c,d / b,d,a /b,d,c
c,a,b /c,a,d / c,b,a / c,b,d / c,d,a / c,d,b
d,a,b / d,a,c / d,b,a/ d,b,c / d,c,a /d,c,b

A Permutation with repetition is the number of ways to arrange elements in a definite order allowing elements to be repeated.
The number of permutations with repetition of n items taken r at a time =

nP(r) (with repetition) = n ^r

Example
The number of permutations of three letters a,b,c arranged two at a time is
3 ² = 9

a,a / a,b / a,c / b,b /b,a / b,c/ c,c /c,a / c,b

A combination is the number of ways to arrange elements with no definite order.  The number of combinations of n elements of same kind taken r at a time.

Equals
n C(r) = n ! / r !(n-r)!

Example.
The number of combinations of four letters a,b,c,d arrange three at a time is
4! / 3!(4-3)! = 4.3.2.1 / 6.1 = 4

a,b,c / a,b,d /a,c,d /b,c,d

Combinations with repetition are the number of ways to arrange elements with no definite order but the elements can be repeated.   The number of combinations of n elements of same kind taken r at a time.

Equals
n C(r) = (n + r -1)! / r ! (n - 1)!

Example.
The number of combinations of four letters a,b,c arrange two at a time is
(3 + 2 - 1)! / (2)!.(3-1)! = 4.3.2.1 / (2.1)(2.1) = 6

a,a /a,b / a,c / b,b / b,c/ c,c /