# Permutations/Arrangements repetitions not allowed

The number of permutations with n things taking r at a time of which a are of one kind, b are of another kind, ... where repetitions are not allowed is given by nuPr

Number of unique items

 = Total number of items − (Number of items of the first kind + Number of items of the second kind + ...)] + Number of kinds of repeated items
 ⇒ nu = [n − (a + b + ...)] + N

# Permutations/Arrangements with items which are all not different

The number of words that can be formed using the letters of a nL letter word taking all at a time (r = n) of which nLa are of one kind, nLb are of another kind, nLc are of a third kind, ...... is given by
 nL! nLa! × nLb! × nLc! × ...

## Example

The number of words that can be formed with the lettes of the word Examinations

In the word Examinations

Number of letters

= 12

{E, X, A, M, I, N, A, T, I, O, N, S}

⇒ nL = 12

Number of letters

of the First Kind (A's) = 2

⇒ nLa = 2

of the Second Kind (I's) = 2

⇒ nLb = 2

of the Third Kind (N's) = 2

⇒ nLc = 2

Number of words that can be formed using all the letters of the word Examinations taking all the letters at a time

=
 nL! nLa! × nLb! × nLc!
=
 12! 2! × 2! × 2!
=
 12 × 11 × 10 × 9! 2 × 1 × 2 × 1 × 2 × 1
= 3 × 11 × 5 × 9!
= 165 × 9!