TITLE: Number of words that can be made using all the letters of the word W, if Os as well as Is are separated is?
QUESTION [8 upvotes]: I am facing difficulty in solving the 18th Question from the above passage. 
Attempt: 
I have attempted it using principal of inclusion and exclusion.
Let n be the required number of ways. 
$n = \text{Total number of ways - Number of ways in which Os and Is are together}$
$\implies n = \dfrac{12!}{3!2!2!} - \dfrac{11!}{2!3!} - \dfrac{10!}{2!2!}+ \dfrac{9!2!}{2!} = 399 \times 8!$
This gives $N = 399$ which is wrong. Can you please tell me why am I getting the  wrong answer?

REPLY [3 votes]: How did you get each of the terms of $n$?

$\frac{12!}{3!2!2!}$ : The number of ways of arranging the letters of SOLICITATION.
$\frac{11!}{2!3!}$ : The number of ways in which the two $O$s can be together in SOLICITATION.
$\frac{10!}{2!2!}$ : The number of ways in which the three $I$s can be together in SOLICITATION.
$\frac{9!2!}{2!}$ :  The number of ways in which the three $I$s and two $O$s can be together in SOLICITATION.

Well , the problem is this : the complement of this case, also includes the case where two $I$s can be together and the third can be separate. You have not neglected this case, and that's why you have an answer larger than the given answer.