How Do You Calculate Permutations of Repeated Letters?

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SUMMARY

The calculation of permutations for the letters in the word "LOLLIPOP" involves accounting for repeated letters. The correct formula is given by \(N=\frac{8!}{3!2!2!}\), where 8 represents the total letters, 3 is for the L's, 2 for the O's, and 2 for the P's. This results in a total of 1680 unique permutations. The initial miscalculation using \(8P8\) failed to consider the identical letters, leading to an incorrect conclusion.

PREREQUISITES
  • Understanding of factorial notation and operations
  • Knowledge of permutations and combinations
  • Familiarity with the concept of identical objects in arrangements
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the principles of permutations with repetition in combinatorics
  • Learn about the factorial function and its applications in probability
  • Explore advanced topics in combinatorial mathematics
  • Practice solving problems involving permutations of words with repeated letters
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Students preparing for mathematics exams, educators teaching combinatorics, and anyone interested in understanding permutations involving repeated elements.

nickar1172
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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?
 
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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.
 
so it would be 8P8/2!3!2! = 1680?
 
Yes, although I would simply write:

$$N=\frac{8!}{3!2!2!}=8\cdot7\cdot6\cdot5=1680$$
 

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