Arranging Letters of a Word (all letters not different)

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Word where all the letters are not different

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The number of permutations with "n" things taking "r" at a time of which "a" are of one kind, "b" are of another kind, "c" are of a third kind, ...... and "x" are all different such that a + b + c + ... + x = n is given by
nPr
a! × b! × c! × ...
(Or)
n! × (n − 1)! × (n − 2)! × ... "r" times
a! × b! × c! × ...

Taking all the letters

The number of words that can be formed using the letters of an "n" letter word taking all at a time ("r" = "n") of which "a" are of one kind, "b" are of another kind, "c" are of a third kind, ...... and "x" are all different such that a + b + c + ... + x = n is given by
nPn
a! × b! × c! × ...
(Or)
n!
a! × b! × c! × ...

Eg: 1. The no. of words that can be formed with the lettes of the word "Examinations"

No. of letters in the word "Examinations" = 12 {E, X, A, M, I, N, A, T, I, O, N, S}

No. of Letters :

of the first kind ⇒ No. of A's = 2 ⇒ a = 2
of the second kind ⇒ No. of I's = 2 ⇒ b = 2
of the third kind ⇒ No. of N's = 2 ⇒ c = 2
which are all different = 6 {E, X, M, T, O, S} ⇒ x = 6

Therefore,
No. of words that can be formed using all (n) the letters of the word "Examinations" taking (r) all the letters at a time

=
n!
a! × b! × c!
=
12!
2! × 2! × 2!
=
12 × 11 × 10 × 9!
2 × 1 × 2 × 1 × 2 × 1
=
3 × 11 × 5 × 9!
1
= 165 × 9!

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