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

\frac{n!}{r_{1}! r_{2}! \dots r_{k}!}

$\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}$

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 $4!$ ways to order the stars and $3!$ ways to order the moon.

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

\frac{n!}{r_{1}! r_{2}! \dots r_{k}!}

$\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}$

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 $n=8, {r}_{1}=2,$ and ${r}_{2}=2$ into the formula.

\frac{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

Counting Principles