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 [latex]12![/latex] 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 [latex]12![/latex] permutations we counted are duplicates. The general formula for this situation is as follows.

[latex]\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}[/latex]

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 [latex]4![/latex] ways to order the stars and [latex]3![/latex] ways to order the moon.

[latex]\frac{12!}{4!3!}=3\text{,}326\text{,}400[/latex]

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 [latex]n[/latex] elements in a set and [latex]{r}_{1}[/latex] are alike, [latex]{r}_{2}[/latex] are alike, [latex]{r}_{3}[/latex] are alike, and so on through [latex]{r}_{k}[/latex], the number of permutations can be found by

[latex]\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}[/latex]

Example 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 [latex]n=8, {r}_{1}=2,[/latex] and [latex]{r}_{2}=2[/latex] into the formula.

[latex]\frac{8!}{2!2!}=10\text{,}080[/latex]

There are 10,080 arrangements.

Try It 10

Find the number of rearrangements of the letters in the word CARRIER.

Solution