MHB How Do You Calculate Permutations of Repeated Letters?

AI Thread Summary
To calculate the permutations of the letters in "LOLLIPOP," the total number of arrangements is derived from the formula N = 8! / (3!2!2!), accounting for the repeated letters: 3 L's, 2 O's, and 2 P's. The correct calculation results in 1680 unique permutations. The initial approach of using 8P8 was incorrect because it did not consider the identical letters. The final expression simplifies to N = 8 × 7 × 6 × 5, confirming the total. Understanding how to adjust for repeated elements is crucial in permutation problems.
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:

$$N=\frac{8!}{3!2!2!}=8\cdot7\cdot6\cdot5=1680$$
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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