Question

The total number of words which can be formed using all the letters of the word "AKSHI" if each word begins with vowel or terminates with vowel

• A. $84$
• B. $12$
• C. $48$
• D. $60$

Answer 1

The correct option is A $84$

Consider the words starting with A.
$A----$
Number of ways of filling the 2nd place is $4$
Number of ways of filling the 3rd place is $3$
Number of ways of filling the 4th place is $2$
Number of ways of filling the 5th place is $1$
Hence in total $4\times 3\times 2\times 1$
$=24$.
Similar cases for the words starting with I.
$=24$.
Hence $2\times 24$
$=48$.
Consider the words ending with A
$----A$
Number of ways of filling the 1st place is $4$
Number of ways of filling the 2nd place is $3$
Number of ways of filling the 3rd place is $2$
Number of ways of filling the 4th place is $1$
Hence in total 24 ways.
Similarly for I- 24 ways.
Hence total number of words $4\times 24$
$=96$.
But this includes the words which start and end with Vowels.
Hence
$A---I$
Number of such words is 3!.
Similarly
$I---A$.
Number of such words is 3!.
Hence total number of words beginning with or ending with Vowels is
$=96-3!-3!$
$=96-12$
$=84$