2. How many distinguishable permutations are possible with all the letters of the word ELLIPSES?
​

5040

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

where n is the total number of objects and  are the number of indistinguishable objects.

We have 2Es, 2Ls, and 2S; the formula then becomes:

Simplifying gives us

There are 5,040 distinguishable permutations of the word ELLIPSES.

For more information about indistinguishable permutations, click here

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents