UPSC Prelims 2007 Q138: Science & Technology – Quantitative Aptitude
All six letters of the name SACHIN are arranged to form different words without repeating
any letter in any one word. The words so formed are then arranged as a dictionary. What will be
the position of the word SACHIN in that sequence?
- (a) 436
- (b) 590
- (c) 601
- (d) 751
Answer: (c) 601
Explanation
Dictionary rank of SACHIN
Letters in alphabetical order: A, C, H, I, N, S.
Each starting letter fixes the first place; the other 5 letters can be arranged in 5! = 120 ways.
| Words starting with | Count |
|---|---|
| A | 120 |
| C | 120 |
| H | 120 |
| I | 120 |
| N | 120 |
| Total before S | 600 |
Now the words starting with S:
The smallest is S + A C H I N = SACHIN itself (remaining letters already in alphabetical order).
So SACHIN is word number 600 + 1 = 601.
Answer: (c) 601.
Why each option is right or wrong
- (a) Incorrect. 436 does not match any correct count.
- (b) Incorrect. 590 is short of the 600 words before 'S'.
- (c) CORRECT. 600 words come before S, and SACHIN is the first S-word → 601.
- (d) Incorrect. 751 over-counts the words that come before SACHIN.
Key point
5 letters come before S → 5 × 5! = 600. SACHIN is the first S-word → rank 601.
Remember
SACHIN is word number 601.
Common mistakes
- Forgetting to add 1 for SACHIN itself (→ 600).
- Sorting letters wrongly — the order is A, C, H, I, N, S.
- Using 6! instead of 5! for each block.
Exam tip
Rank = (words before) + 1. Count blocks of 5! for each smaller starting letter.
Sources
- NCERT Class XI — Mathematics, Ch 6 (Permutations and Combinations)
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)