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Trying to solve a problem from Brilliant: how many ways are there to arrange the letters in the word "FLOOD" such that the two O's are not next to each other?

My approach: 3 ways to split the two O's (one for each consonant), times 3 ways to position the O[FLD]O group with respect to the two remaining letters (left, middle, right), times 2 ways to put the other two letters (ex. OFOLD and OFODL).
Result: 3*3*2 = 18 (wrong)

I understand that the correct result can be obtained calculating the total possible combinations (5! divided by 2: contiguous O's are not to be counted twice) minus the total of F, L, D, and OO arrangements (4!).
Correct result: 5!/2 - 4! = 36

What's missing from my approach?

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There are more than 1 letter can be in two 'O' ?

I think figure out the answer /difference between your solution and correct solution is not hard. Just manually list all of the and see what cases been missed.

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  • $\begingroup$ Of course... I completely missed that possibility. Thanks $\endgroup$
    – Dario R
    Commented Apr 14, 2021 at 10:20

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