vabhs192003 wrote:Hello,

I have a conceptual problem in application in the below question.

How many words can be made from the word IMPORTANT,using all the 9 alphabets, in which both the Ts do not come together, ?
a) 141130
b) 141210
c) 113311
d) 888222
e) 141120
OA: "e".

Here's one approach.

First ignore the restriction that says the T's cannot be adjacent to one another. So, we have 9 objects (letters) and we want arrange all of them.

Note: When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:

If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....]

In this question, the word IMPORTANT has two identical letters (the T's), so when we apply the MISSISSIPI rule, we see that the letters in IMPORTANT can be arranged in 9!/2! ways.

Now at this point, we've ignored the restriction that says the T's cannot be adjacent to one another.

So, from the 9!/2! possible words, we need to subtract all of the words in which the T's are adjacent. How many such words are there?

To find out, let's "glue" the two T's together, to get one symbol "TT." This will ensure that the letters are adjacent.
So, we want to determine how many ways we can arrange the following 8 "letters": I, M, P, O, R, A, N, TT
This can be accomplished in 8! ways. Notice that all 8! words will have the two T's adjacent.

So, the total number of words in which the T's are not adjacent = 9!/2! - 8! = [spoiler]141,120 = E[/spoiler]

Cheers,
Brent