SOLUTION: How many ways can the letter in the word OPTIMISM be rearranged? What I did: 2 Is, 2 Ms, 1 O, 1 P, 1 T, 1 S = (8,2)(6,2)(4,1)(3,1)(2,1)(1,1) {{{ (8!/2!6!)(6!/2!4!)(4!/1!3!)

I'm afraid not.

OPTIMISM is an 8-letter word.  If the word were "oPTimISM" the answer would be
8! =40320 because we would be able to tell the difference between the arrangements

mSiToPIM, mSIToPiM, MSIToPIm, MSiToPIm

But since we cannot because the I's and M's look the same, we must divide the
9! by the number of ways the I's can be swapped, which is 2! and also by the
number of ways the M's can be swapped, which is also 2!.

Answer =  = 10080

You see, we had to divide by 2!2! = 4 because, as we saw with the four arrangements

mSiToPIM, mSIToPiM, MSIToPIm, MSiToPIm, all look the same as MSITOPIM

every arrangement is counted 4 times too many in the 8!, that's why we divide
by 4.  We divide by the factorial of the number of times each letter is
repeated.

Edwin