How Do You Calculate Permutations of Repeated Letters?

[nickar1172]

Reviewing for finals and got this question wrong:

How many different permutations are there of the letters in the word LOLLIPOP

what I did was 8P8, how would you solve this?

 

[MarkFL]

The number of ways to order or arrange $n$ objects is $n!$. So, we want to look at the number of ways to order 8 letters, however, there are 3 L's, 2 O's and 2 P's. Hence, you want to take the total number of ways to arrange 8 letters, and then account for the fact that some of them are identical. Can you state how many would be identical?

edit: I have removed the [SOLVED] label from the title so that our readers don't skip the thread thinking you have already found the solution yet.

 

[nickar1172]

so it would be 8P8/2!3!2! = 1680?

 

[MarkFL]

Yes, although I would simply write:

\(\displaystyle N=\frac{8!}{3!2!2!}=8\cdot7\cdot6\cdot5=1680\)

 

[CellCoree]

Your method of using 8P8 is correct. This is because there are 8 letters in the word LOLLIPOP and you are finding the number of ways to arrange all 8 letters without repetition. This is known as a permutation because the order of the letters matters.

Another way to solve this would be to use the formula for permutations of n objects taken r at a time, where n is the total number of objects and r is the number of objects being selected. In this case, n = 8 and r = 8, so the formula would be nPr = 8P8 = 8!/(8-8)! = 8!/0! = 8! = 40320.

In summary, there are 40320 different permutations of the letters in the word LOLLIPOP.