Otdfbt

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.

$n = textTotal number of ways - Number of ways in which Os and Is are together$

$implies n = dfrac12!3!2!2! - dfrac11!2!3! - dfrac10!2!2!+ dfrac9!2!2! = 399 times 8!$

This gives $N = 399$ which is wrong. Can you please tell me why am I getting the wrong answer?

Your mistake is not noticing that the I's can be non-separated two different ways: We can have all three of them together as III, but we can also have II in one place and I in another.

One way to calculate how many ways this can happen is to think of one pair II as a single letter, and the third I as a separate letter, and count the number of words we can make. This will count each III case twice: Once as I II, and once as II I. So we have to subtract the number of such cases once.

With this correction, the final calculation becomes
$$
overbracefrac12!3!2!2!^textAll words - overbracefrac11!3!2!^textO's not separate - overbraceleft(underbracefrac11!2!2!_textII and I - underbracefrac10!2!2!_textIIIright)^textI's not separate + overbraceleft(underbracefrac10!2!_textOO, II and I - underbracefrac9!2!_textOO and IIIright)^textNeither O's nor I's separate
$$
which turns out to be $8!cdot 228$.

The letter $I$ appears three times. You have calculated the number of ways to arrange the letters so that the three copies of $I$ aren't all in the same place (and also the two $O$s are separated). However, "separated" here means that no two of them are together. Your count includes such "words" as STIILOCATINO, which should be excluded.

How did you get each of the terms of $n$?

• $frac12!3!2!2!$ : The number of ways of arranging the letters of SOLICITATION.




• $frac11!2!3!$ : The number of ways in which the two $O$s can be together in SOLICITATION.




• $frac10!2!2!$ : The number of ways in which the three $I$s can be together in SOLICITATION.




• $frac9!2!2!$ : 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.