Arranging Letters of a Word (all letters not different)
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
= |
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= |
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= | 3 × 11 × 5 × 9! | ||
= | 165 × 9! |
Author : The Edifier