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here are 4 kinds of letters: 1 M, 2 Ps, 4 Ss and 4 Is available to form a 4-letter word. But we must not get confounded with the count of each kind of letters rather we must only consider the 4 kinds of letters: M, P, S and I.
We can break down the number of 4 letter words that can be formed into sub-categories:
1. Words having all distinct letters
2. Words with exactly one letter repeated twice
3. Words with exactly one letter repeated thrice
4. Words with all letters of same kind
5. Words with two distinct letters, each repeated twice.
Let us deal with each cases separately.
1. There are exactly 4 different kinds of letters: M,I,S and P. So, the total number of words with all distinct letters = 4! = 24
2. There are three different kinds of letters that can be repeated and they are I, P and S.
The letter that has to be repeated can be selected in 3 ways.
The two distinct letters can be selected in 3C2 =3ways
Also, each combination can be arranged in 4!2! =12ways
So, the number of words with exactly one letter repeated twice = 12 ∗ 3 ∗ 3C2 =108 ways
3. There are two different kinds of letters that can be used three times and they are I and S.
The letter to be repeated thrice can be selected in 2C1 ways
The remaining one letter can be selected in 3C1 =3 ways
Also, each combination can be arranged in 4!3! =4ways
So, the number of words with exactly one letter repeated thrice = 4 ∗ 2C1 ∗ 3C1 =24
4. The letter could be either I or S. So, there are only 2 words: IIII and SSSS, in this case.
5. Two doubly repeated letters can be selected from P , I and S in 3C2 =3 ways
Each combination can be rearranged in 4!2!∗2! =6 ways
So, the total number of words with doubly repeated letters of two kinds = 3*6 = 18
Therefore, the total number of 4 letter words = 24 + 108 + 24 + 2 + 18 = 176
Regards
Arun (askIITians forum expert)
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