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
Best Answer
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.