How many different 5-letter words (with or without meaning) can be constructed using all the letters of the word 'DELHI' so that each word has to start with D and end with I?
- (a) 24
- (b) 18
- (c) 12
- (d) 6 ✓ UPSC's answer
Why the answer is (d)
• The word 'DELHI' contains 5 distinct letters: D, E, L, H, and I.
• Each required 5-letter word must begin with 'D' and end with 'I', which means the 1st position and the 5th position are fixed.
• This leaves the 2nd, 3rd, and 4th positions to be filled using the remaining 3 letters: E, L, and H.
• The number of ways to arrange these 3 distinct letters in the 3 available positions is given by 3! (3 factorial).
• Evaluating the factorial gives 3 × 2 × 1 = 6 distinct words, matching option (d).
Why the other options are wrong
- (a) 24
- This corresponds to 4! (24), which represents the number of arrangements if only one letter were fixed instead of both 'D' and 'I'.
- (b) 18
- This is an incorrect calculation, possibly resulting from incorrectly multiplying the 6 arrangements by 3.
- (c) 12
- This value would occur if there were 2 independent ways to place the remaining letters, or from confusing 3! with 2 × 3!.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.