Question

Number of words that can be made with the letters of the word “GENIUS” if each word neither begins with G and nor ends with S is:
(A) $ 24 $
(B) $ 240 $
(C) $ 480 $
(D) $ 504 $

Answer 1

Hint: In the given question, we are given the word “GENIUS” and we have to determine how many words can be made with the letters of the word given such that any one of the words neither begins with G nor ends with S. The given questions involve the concepts of permutations and combinations. One should know the inclusion and exclusion principle of the permutations and combinations in order to solve this problem.

Complete step by step solution:
So, we have the word “GENIUS”.
So, we have to make sure that no word starting with G nor ending with S is counted.
Firstly, we count all the words that can be formed by the letters of the word “GENIUS”.
So, the number of letters of the word “GENIUS” $ = 6 $
So, the number of words that can be formed with the letters of the word “GENIUS” is $ 6! $ .
So, we can form a total of $ 720 $ words with the letters of the word “GENIUS”.
Now, we count the words starting with the letter G.
So, there are only five spots that need to be filled. Hence, the number of words starting with G is $ 5! $ .
So, we can form a total of $ 120 $ words with the letters of the word “GENIUS” starting with G.
Similarly, we find out the number of words ending with S.
So, there are only five spots that need to be filled. Hence, the number of words ending with S is $ 5! $ .
So, we can form a total of $ 120 $ words with the letters of the word “GENIUS” ending with S.
Now, we have to find out the number of words starting with G and ending with S.
So, there are only four spots that need to be filled. Hence, the number of words ending with S is $ 4! $ .
So, we can form a total of $ 24 $ words with the letters of the word “GENIUS” ending with S and starting with G.
Now, we can find the number of words that neither begins with G and nor ends with S using the principle of inclusion and exclusion.
So, the number of words that neither begins with G and nor ends with S can be calculated by subtracting the number of words that end with S and words that start with G from the total number of words that can be formed. But in the process, the words that both start with G and end with S are subtracted twice. So, we add the number back one time.
So, we get, $ \left( {720 - 120 - 120 + 24} \right) $ as the number of words that neither end with S nor start with G.
Thus, we get, $ 504 $ is the final answer to the question.
Hence, option D is correct.
So, the correct answer is “Option D”.

Note: The principle of inclusion and exclusion is a counting technique that computes the number of elements that satisfy at least one of several properties while guaranteeing that elements satisfying more than one property are not counted twice.