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12 letter words have to be created using 3 Ps, 4 Qs and 5 Rs such that no two Rs are together.
To start, create words that have only P and Qs. The number of such words that can be created is merely the number of ways of choosing 3 spaces out of a total of 8 where P is filled in; the rest are filled by the letter Q. This gives the number of letters as C(8, 3) = 56.
The Rs have to be arranged such that no two of them are together. To do this the Rs are filled in spaces between the Ps and Qs. There are 9 such spaces and a total of 5 Rs to fill in. This can be done in C(9, 5) = 126 ways.
The total number of 12 letter words that can be formed is 56*126 = 7056.
There are 7056 12-letter words that can be created using 3 Ps, 4 Qs and 5 Rs such that no two Rs are together.
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