Example 3d (Sheldon Ross) on Arrangement of Letters

How many different letter arrangements can be formed from the letters PEPPER?

Answer

We first note that

There are 6! permutations of the letters P1E1P2P3E2R when the 3P’s and the 2E’s are distinguished from each other.

However, consider any one of these permutations—for instance, P1P2E1P3E2R. If we now permute the P’s among themselves and the E’s among themselves, then the resultant arrangement would still be of the form PPEPER. That is, all 3! 2! permutations

P1P2E1P3E2R     P1P2E2P3E1R
P1P3E1P2E2R     P1P3E2P2E1R
P2P1E1P3E2R     P2P1E2P3E1R
P2P3E1P1E2R     P2P3E2P1E1R
P3P1E1P2E2R     P3P1E2P2E1R
P3P2E1P1E2R     P3P2E2P1E1R

are of the form PPEPER. Hence, there are \({6! \over (3! 2!)} = 60\) possible letter arrangements of the letters PEPPER. 

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