Question 717781: Q4. No. of words that can be formed using all the letters of the word GARGEE
if no two alike letters are together.
Ans is 84.
Answer by Positive_EV(69)   (Show Source):
You can put this solution on YOUR website! This is a somewhat more tricky problem than it looks. The first step is to find the number of total orderings of the letters, ignoring the restriction that like letters cannot be placed together. This value is 6!/((2!)(2!)(1!)(1!)) = 180.
Now, if you consider the cases that involve like letters together, you can consider a pair of like letters as one "unit" -- that is, since the like letters are together, you can treat them as a single object for the calculations. The number of permutations with the two G's in a row is 5!/((2!)(1!)(1!)(1!)) = 60. The number of permutations with the two E's in a row is exactly the same: 5!/((2!)(1!)(1!)(1!)) = 60.
Now, there's a problem. 180 - 60 - 60 = 60, not 84. The last trick to the problem is that the sets of permutations with one pair of similar letters together will also have certain permutations will the other set of letters together as well. The problem is that, in performing these calculations, we've counted the arrangements with both sets of like letters together twice. Thus, by the inclusion-exclusion theorem, we have to add back a set of the permutations with both sets of alike letters together. In this case, we treat both pairs as a single unit each, so there's only 4 units of letters (2 pairs of like letters and the two single letters) to put in a line. There are 4! = 24 ways to arrange the letters so both pairs of similar letters are together. Thus, since we counted these patterns twice when subtracting the "one set of like letters together" permutations, we have to add back one set of these to prevent counting them twice.
Thus, the final outcome is (number of total permutations) - (number of permutations with the G's together) - (number of permutations with the E's together) + (number of permutations with both the G's and E's together) = 180 - 60 - 60 + 24 = 84.
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