permutations of the letters of the word MISSISSIPPI without occurrence of IIII Total no. of permutations of the letters of the words $\bf{MISSISSIPPI}$ in which no four $\bf{I,s}$ come together.
My Try:: (Using Complementry Counting)
Total no. of the permutations of the words is $\displaystyle =\frac{11!}{4! \times 4! \times 2!}$
Now Total no. of permutations of the word, in which four $\bf{I,s}$ come together $\displaystyle  = \frac{8!}{4! \times 2!}$.
Now Total permutatations in which $\bf{4-I,s}$ come together, is $\displaystyle = \frac{11!}{4! \times 4! \times 2!}-\frac{8!}{4! \times 2!}$
Now my Question is Can we solve it  without Using  Complementry Counting,
If Yes , Then How Can we solve it.
Thanks  
 A: Broadly, the answer is yes we can. The more important question is, should we bother? There are lots of ways you could do it - just enumerate every single arrangement, or work out what "templates" fit the requirements and then calculate the permutations for each template, but complementary counting gives you the answer in quite a simple form.
A: First note that except for the basic problems, which are solved in order students to get used to combinatorics and to get a routine, general formulas doesn't exist, because combinatorics is such a diverse field in the paths.
So you need to apply logic and creativity to find all combinations and possibilites and then to find a generating formula for every one of them.
So I don't think that exist a particular formula for this problem, but you can made one for you. This would have been a classic stars and bars problem if it wasn't allowed two $I's$ to be together. In this problem having 3 $I$ together is allowed, as long as the fourth $I$ isn't next to them, but stars and bars wouldn't allow that.
I think your reasoning is the best, simpliest and easiest solution to this problem. First find all posiblle combinations and then find the number of "bad" combinations and at the end subtact them from the total number of combinations in order to get the final answer.
