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`        How many arrangements can be made by taking 4 letters of the word Mississippi?`
2 years ago

Arun
23742 Points
```							Dear Sohan 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 letters2. Words with exactly one letter repeated twice3. Words with exactly one letter repeated thrice4. Words with all letters of same kind5. 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! = 242. 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 =3waysAlso, each combination can be arranged in 4!2! =12waysSo, the number of words with exactly one letter repeated twice = 12 ∗ 3 ∗ 3C2 =108 ways3. 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 waysThe remaining one letter can be selected in 3C1 =3 waysAlso, each combination can be arranged in 4!3! =4waysSo, the number of words with exactly one letter repeated thrice = 4 ∗ 2C1 ∗ 3C1 =244. 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 waysEach combination can be rearranged in 4!2!∗2! =6 waysSo, the total number of words with doubly repeated letters of two kinds = 3*6 = 18Therefore, the total number of 4 letter words = 24 + 108 + 24 + 2 + 18 = 176 RegardsArun (askIITians forum expert)
```
2 years ago
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• Mind Map
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