# 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

^{nu}P_{r}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 |

⇒ n_{u} | = | [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 n

_{L}letter word taking all at a time (r = n) of which n_{La}are of one kind, n_{Lb}are of another kind, n_{Lc}are of a third kind, ...... is given by n_{L}! |

n_{La}! × n_{Lb}! × n_{Lc}! × ... |

## 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}

⇒ n_{L} = 12

Number of letters

of the First Kind (A's) = 2

⇒ n_{La} = 2

of the Second Kind (I's) = 2

⇒ n_{Lb} = 2

of the Third Kind (N's) = 2

⇒ n_{Lc} = 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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= |
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= | 3 × 11 × 5 × 9! | ||

= | 165 × 9! |

Author : The Edifier