Rank of a Word in Dictionary: Formula, Tricks and Solved Questions for SSC Reasoning

The question "If all the letters of the word MATH are arranged in all possible ways and written as in a dictionary, what is the rank of MATH?" is asked in some SSC and bank exams. There is a simple formula that gives the rank in a minute for words with different letters. These rank of a word in dictionary notes for SSC explain the steps and solve the examples.

Idea

The words are written in dictionary (alphabetical) order. All the words that start with a smaller letter come first, and so on. The number of ways to arrange n different letters is n! (for example 3! = 6 and 4! = 24).

Steps (Words With Different Letters)

1. Take the first letter. Count how many letters on its right are smaller (earlier in the alphabet).
2. Multiply this count by (n − 1)!, where n is the number of letters.
3. Go to the second letter. Count the smaller letters on its right, and multiply by (n − 2)!.
4. Do the same for each letter until the last.
5. Add all the numbers and add 1. This is the rank.

Factorials

Solved Examples

Example 1. Find the rank of CAT.
C: smaller letters on the right are A (1) × 2! = 2. A: none, 0. T: none, 0. Rank = 2 + 1 = 3. (The order is ACT, ATC, CAT, CTA, TAC, TCA.)

Example 2. Find the rank of DOG.
D: none smaller on the right (G and O are greater), 0. O: G is smaller, 1 × 1! = 1. G: 0. Rank = 1 + 1 = 2.

Example 3. Find the rank of MATH.
M: smaller letters on the right are A and H (2) × 3! = 12. A: none, 0. T: H is smaller, 1 × 1! = 1. H: 0. Rank = 12 + 1 + 1 = 14.

Example 4. Find the rank of SAFE.
S: A, F and E are smaller (3) × 3! = 18. A: 0. F: E is smaller, 1 × 1! = 1. E: 0. Rank = 18 + 1 + 1 = 20.

Example 5. Find the rank of CBA.
C: B and A are smaller (2) × 2! = 4. B: A is smaller (1) × 1! = 1. A: 0. Rank = 4 + 1 + 1 = 6, the last word.

Tips

• The word in alphabetical order (like ABC or ACT) always has the rank 1.
• The word in reverse alphabetical order (like CBA) has the last rank, n!.
• If a letter repeats, divide the result by the factorial of the repeat count (this is asked rarely).
• Write the factorial values at the top of your rough sheet.

Quick Tips

• Count the smaller letters to the right of each letter.
• Multiply the count by the factorial of the places that are left.
• Add 1 at the end.
• The first word in alphabetical order has the rank 1.

Quick Revision Points

• Rank = sum of (smaller letters on the right × factorial) + 1.
• 4! = 24 and 3! = 6.
• Alphabetical order gives rank 1.
• Reverse order gives the last rank n!.
• CAT is at the rank 3.

Practice Questions With Answers

1. Find the rank of CAT.
   Answer: 3
2. Find the rank of DOG.
   Answer: 2
3. Find the rank of BAT.
   Answer: 3
4. Find the rank of CBA.
   Answer: 6
5. Find the rank of MATH.
   Answer: 14
6. Find the rank of SAFE.
   Answer: 20
7. Find the rank of ABC.
   Answer: 1
8. Find the rank of TAC.
   Answer: 5
9. In how many ways can the letters of MATH be arranged?
   Answer: 24
10. In how many ways can the letters of DOG be arranged?
    Answer: 6

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Verbal Reasoning notes page, read the previous topic Vowels, Consonants and Letter Position and the next topic Letter Group Classification. Learn the letter positions from the Alphabet Test note first. Also read Dictionary Order and Word Arrangement.

Frequently Asked Questions

What is the rank of a word in a dictionary?

It is the place of the word when all the arrangements of its letters are written in dictionary order.

What is the formula of the rank?

The sum of (smaller letters on the right of each letter × factorial of the remaining places) plus 1.

How many arrangements can n different letters make?

n factorial (n!).

What is the rank of the word that is in alphabetical order?

The rank is 1.

What is the rank of the word in reverse alphabetical order?

The rank is n!, the last one.

What is the rank of CAT?

The rank is 3.

What is the rank of MATH?

The rank is 14.

What if letters repeat?

Divide the number of arrangements by the factorial of the repeated letters.

Is this topic asked in SSC exams?

It is asked in some SSC and bank exams.

How much time should I take?

About 60 seconds.

What are the common mistakes?

Counting the smaller letters on the left and forgetting to add 1.

What are factorials?

n! is the product of all the numbers from 1 to n.