TITLE: Arranging the word 'MISSISSIPPI'
QUESTION [5 upvotes]: "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.

REPLY [3 votes]: The number of solutions to $x_1+x_2+x_3+x_4+x_5 = 7$ with $0 \leq x_i \leq 2$ is same as the coefficient of $x^7$ in $(1+x+x^2)^5$. This can be expanded using binomial theorem as $(1+x(1+x))^5$. Since $x^7$ will occur only in the terms $\binom{5}{4}x^4(1+x)^4$ and $\binom{5}{5}x^5(1+x)^5$, the required coefficient is $\binom{5}{4}\binom{4}{3} + \binom{5}{5}\binom{5}{2} = 30$. Since there are $\binom{7}{4,2,1} = 105$ ways to place $SSSSPPM$ in these 7 places, the required answer is 3150.