"How many ways are there to arrange the letters in the word 'MISSISSIPPI' in such a way that there are no three consonants in a row?"
I am thinking like this. The following are 'slots' for the letters of our word: _ _ _ _ _ _ _ _ _ _ _. There are 21 consonants in total (English). The first slot as 21 possibilities, the second also has 21, but the third has only 5 possibilities. So how many ways can I place vowels within this word... I don't know.
I can try to enumerate all of the place the I's can go, but there has to be a better way than doing that.