Answer:
5,040
Step-by-step explanation:
This problem is about indistinguishable permutation. We have indistinguishable objects, in this case, these are the letter of the word ELLIPSES. The permutations for the word ELLIPSES is the same when you swap the places of the Ls. We need to not count them to avoid double counting.
We first count how many letters we have on the word, and then count the repeating letters.
ELLIPSES has 8 letters, 2 Es, 2Ls, and 2S.
The formula for indistinguishable permutation is
[tex]\frac{n!}{n_{1}!n_{2}!...n_{k}! }[/tex]
where n is the total number of objects and [tex]n_{k}[/tex] are the number of indistinguishable objects.
We have 2Es, 2Ls, and 2S; the formula then becomes:
[tex]= \frac{8!}{2!2!2!}\\[/tex]
Simplifying gives us
[tex]= \frac{8!}{2!2!2!}\\\\= \frac{8*7*6*5*4*3*2*1}{2*2*2}\\\\= 8*7*6*5*3\\\\= 5,040[/tex]
There are 5,040 distinguishable permutations of the word ELLIPSES.
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