Distinguishable Arrangments

Question

What is the number of distinguishable arrangements of the letters of the word SEXTILLION

Answer 1

SEXTILLION has 10 letters.

If you had 10 different letters then you could arrange them in 10! different ways (order matters).

But in SEXTILLION we have 2 I's and 2 L's. For the I's, or the L's, we don't care about their order.

One of the possible arrangements is II........ where the first I is first and the second I is second, but in 10! we counted II....... again where the second I is first, and the first I is second. Basically we double counted. Same for the 2 L's. To account for this we need to divide by 2 twice

10! / 2*2

For fun try to find the total number of distinguishable arrangements of the letters of MISSISSIPPI