Problem Solving : Arranging Letters of a Word (all letters are not different)

Permutations/Arrangements with repetitions

The number of permutations or arrangements 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! × ...

The product a! × b! × c! × ... should not contain the value of 'x'.

Words formed by taking all the letters of a word (all letters are not different)

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! × ...

Illustrations

1. The number of words that can be formed with the lettes of the word "examinations"

Solution Show

In the given word

L = Total number of letters
= 12 {E, X, A, M, I, N, A, T, I, O, N, S}
La = Repeating letters of the first kind
= 2 {A's}
Lb = Repeating letters of the second kind
= 2 {I's}
Lc = Repeating letters of the third kind
= 2 {N's}
Lx = Letters without repetitions
= 6 {E, X, M, T, O, S}
L = La + Lb + Lc + Lx

In the word to be formed

P = Number of places to be filled to form the word
= 12

The number of words that can be formed using all the letters of the word "examinations" taking all the letters at a time

=
L!
La! × Lb! × Lc!
=
12!
2! × 2! × 2!
=
12 × 11 × 10 × 9!
2 × 1 × 2 × 1 × 2 × 1
=
3 × 11 × 5 × 9!
1
= 165 × 9!

... 141516 ...