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Study Guides > College Algebra

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

n!r1!r2!rk!\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}
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 4!4! ways to order the stars and 3!3! ways to order the moon.
12!4!3!=3,326,400\frac{12!}{4!3!}=3\text{,}326\text{,}400
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 nn elements in a set and r1{r}_{1} are alike, r2{r}_{2} are alike, r3{r}_{3} are alike, and so on through rk{r}_{k}, the number of permutations can be found by
n!r1!r2!rk!\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}

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 n=8,r1=2,n=8, {r}_{1}=2, and r2=2 {r}_{2}=2 into the formula.
8!2!2!=10,080\frac{8!}{2!2!}=10\text{,}080
There are 10,080 arrangements.

Try It 10

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

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  • Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.