Finding the Number of Permutations of n Non-Distinct Objects
We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the permutations we counted are duplicates. The general formula for this situation is as follows.
In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are ways to order the stars and ways to order the moon.
There are 3,326,400 ways to order the sheet of stickers.
A General Note: Formula for Finding the Number of Permutations of n Non-Distinct Objects
If there are elements in a set and are alike, are alike, are alike, and so on through , the number of permutations can be found byExample 6: Finding the Number of Permutations of n Non-Distinct Objects
Find the number of rearrangements of the letters in the word DISTINCT.Solution
There are 8 letters. Both I and T are repeated 2 times. Substitute and into the formula.
There are 10,080 arrangements.