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

Solution

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 $$=96-3!-3!$$$$=96-12$$$$=84$$Maths

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