Question

Based on this information answer the questions given below.
A string of three English letters is formed as per the following rules:
(a) The first letter is any vowel.
(b) The second letter is $m,n$ or $p$.
(c) If the second letter is $m$ then the third letter is any vowel which is different from the first letter.
(d) If the second letter is $n$ then the third letter is $e$ or $u$.
(e) If the second letter is $p$ then the third letter is the same as the first letter.

How many strings of letters can possibly be formed using the above rules?

• A. $40$
• B. $45$
• C. $30$
• D. $35$

Answer 1

Answer: (D)

$35$

There are exclusive cases:
(i) when m is the second letter(ii) when n is the second letter(i) when p is the second letter
Case (i): First letter can be selected in 5 ways out of t vowels. Since second letter is fixed (i.e., m) therefore no. of
ways of selection of second letter is 1.
Third letter can be selected in 4 ways out of remaining 4 vowels (since the vowel which has been used at first place can
not be used at the place of third letter.
$\therefore$ Total no. of ways $=5\times 1\times 4=20$

Case (ii): First letter can be selected in 5 ways.Second letter can be selected in 1 way.Third letter can be selected in 1 way.Since third letter will be same as the first letter. So if the first letter is selected, then there is no need to select
the third letter.
$\therefore$ Total no. of ways $=5\times 1\times 1=5$
Hence , sum of all the possible no. of ways in which the string of letters can be formed
$=20+10+5=35$