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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
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B
12
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C
48
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D
60
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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
$$=96-3!-3!$$
$$=96-12$$
$$=84$$

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