Number of words if each word neither begins with G nor ends with S

Question

Number of words that can be made with the letters of the word " GENIUS " if each word neither begins with G nor ends with S.

$(a)$ $24$

$(b)$ $240$

$(c)$ $480$

$(d)$ $504$

Are we supposed to make cases here like two letter word, three letter word and so on?

Is there any easier approach to this problem?

Answer 1

Use inclusion/exclusion principle:

• Include the total number of words, which is $6!$
• Exclude the number of words starting with G, which is $5!$
• Exclude the number of words ending with S, which is $5!$
• Include the number of words starting with G and ending with S, which is $4!$

Hence the number of such words is $6!-5!-5!+4!=504$.