Finding the Number of Permutations of [latex]n[/latex] Non-Distinct Objects
We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable?
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However, [latex]4[/latex] of the stickers are identical stars, and [latex]3[/latex] are identical moons. Because all of the objects are not distinct, many of the [latex]12![/latex] permutations we counted are duplicates.
[latex]\\[/latex]
This means we need to divide by the number of ways to order the [latex]4[/latex] stars and the ways to order the [latex]3[/latex] 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]\dfrac{12!}{4!3!}=3\text{,}326\text{,}400[/latex]
There are [latex]3,326,400[/latex] ways to order the sheet of stickers.
formula for finding the number of permutations of [latex]n[/latex] 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]\dfrac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}[/latex]