Question

How many ways are there to arrange the 26 letters of the alphabet in a row such that none of the following words are formed by consecutive letters in the arrangement: INCH, LOST, or THIN?

Answer is 26! - 3x23! + 2x20!, please explain how to get it, thanks.

Answer 1

Answer 2

To solve this problem, we need to use the principle of inclusion-exclusion. We first count the total number of ways to arrange the 26 letters of the alphabet in a row, which is simply 26!. However, this count includes arrangements that form the forbidden words as consecutive letters.

Next, we count the number of arrangements that form the word INCH as consecutive letters. We can treat the four letters of INCH as a block and arrange the remaining 22 letters in a row. This can be done in 23! ways. Similarly, we can count the number of arrangements that form the word LOST or THIN as consecutive letters. For LOST, we can treat the four letters as a block and arrange the remaining 22 letters in a row in 23! ways. For THIN, we can treat the three letters as a block and arrange the remaining 23 letters in a row in 23! ways.

However, we need to be careful because some arrangements may form more than one forbidden word as consecutive letters. For example, the arrangement INCHLOST... forms both INCH and LOST as consecutive letters. To correct for this, we need to subtract the number of arrangements that form two of the forbidden words and add back the number of arrangements that form all three forbidden words as consecutive letters.

There are 2 ways to form two of the forbidden words (INCH and LOST), and for each of these, we can treat the six letters as a block and arrange the remaining 20 letters in a row in 20! ways. There is only one way to form all three forbidden words (INCHLOST), and we can treat the seven letters as a block and arrange the remaining 19 letters in a row in 19! ways.

Therefore, the number of arrangements that do not form any of the forbidden words as consecutive letters is:

26! - (3 x 23!) + (2 x 20!) - (2 x 20!) + 19! = 26! - 3 x 23! + 2 x 20!

Hence, there are 26! - 3x23! + 2x20! ways to arrange the 26 letters of the alphabet in a row such that none of the words INCH, LOST, or THIN are formed by consecutive letters in the arrangement.