Rank of a Word in the Dictionary: The 60-Second Trick
If every arrangement of the letters of MOTHER were printed in a dictionary, which page would MOTHER land on? Here's how to answer it without listing all 720 words.
What does "rank of a word" mean?
Take the letters of a word and write every possible arrangement of them, whether it means anything or not. Then sort that list in alphabetical (dictionary) order. The rank of a word in the dictionary is simply the position where the original word appears in that list.
This is one of the most repeated questions in permutation and combination for JEE Main, board exams and NCERT Class 11 Maths (Chapter 6). It looks scary because the lists are huge, but you never need to write the list. You only need to count the words that come before.
See it with a tiny word first
The word CAT has \(3! = 6\) arrangements. Press the button and watch the "dictionary" being read from the top.
For CAT, the words ACT and ATC start with A, which is smaller than C. So 2 words come before CAT, and its rank is 3. That's the whole method; the rest is doing it for longer words.
Example 1: rank of MOTHER = 309
MOTHER has 6 different letters, so there are \(6! = 720\) arrangements. First, write the letters in alphabetical order:
E H M O R T
Now go through MOTHER one letter at a time. At each position, ask: "Which unused letters are smaller than this one?" Every word that puts a smaller letter here comes earlier in the dictionary, and the remaining letters can be arranged freely after it.
| Position | Letter | Unused smaller letters | Words before |
|---|---|---|---|
| 1st | M | E, H (2 letters) | \(2 \times 5! = 240\) |
| 2nd | O | E, H (M is used) | \(2 \times 4! = 48\) |
| 3rd | T | E, H, R | \(3 \times 3! = 18\) |
| 4th | H | E | \(1 \times 2! = 2\) |
| 5th | E | none | 0 |
| 6th | R | none | 0 |
| Total words before MOTHER | 308 | ||
Example 2: rank of SUCCESS = 331 (repeated letters)
SUCCESS has 7 letters, but C appears 2 times and S appears 3 times. Swapping two identical S's gives the same word, so we must not count it twice. The number of distinct arrangements is:
Alphabetical order of the distinct letters: C E S U. The rule now changes slightly: for each smaller letter you try, remove one copy of it, then count arrangements of what's left using the factorial formula with repeats.
| Pos. | Letter | Try a smaller letter | Letters left | Words before |
|---|---|---|---|---|
| 1st | S | C | C, E, S, S, S, U | \(\dfrac{6!}{3!} = 120\) |
| E | C, C, S, S, S, U | \(\dfrac{6!}{2!\,3!} = 60\) | ||
| 2nd | U | C | C, E, S, S | \(\dfrac{5!}{2!} = 60\) |
| E | C, C, S, S | \(\dfrac{5!}{2!\,2!} = 30\) | ||
| S | C, C, E, S | \(\dfrac{5!}{2!} = 60\) | ||
| 3rd–7th | C C E S S | already in order | 0 | |
| Total words before SUCCESS | 330 | |||
The general formula
For a word of length \(n\), at each position \(i\), try every unused letter \(x\) that is smaller than the letter in that position. Remove one copy of \(x\), and let \(r_1, r_2, \dots\) be how many times each letter repeats in what is left:
When all letters are different, every \(r = 1\), so it becomes the simpler version you used for MOTHER, where \(s_i\) is the number of unused letters smaller than the \(i\)-th letter:
Try any word: rank calculator
Letters A to Z only, up to 15 letters. Try LETTER, INDIA or your own name.
Watch the two Shorts
MOTHER → 309 (all letters different)
SUCCESS → 331 (with repeated letters)
More solved examples
| Word | Total arrangements | Rank | Quick reason |
|---|---|---|---|
| CAT | 6 | 3 | ACT, ATC come first |
| BOOK | 12 | 3 | \(\dfrac{4!}{2!} = 12\); only BKOO and BOKO come before |
| APPLE | 60 | 12 | A is smallest, so count only from position 2 |
| INDIA | 60 | 46 | I repeats twice |
| MATHS | 120 | 53 | \(2\cdot 4! + 2\cdot 2! = 52\), plus 1 |
| LETTER | 180 | 71 | E and T both repeat |
Every rank in this table was checked by listing all arrangements with a computer program.
Practice questions (tap to see the answer)
1. Find the rank of LEARN.
56. Letters A E L N R: L gives \(2 \times 4! = 48\), E gives \(1 \times 3! = 6\), A gives 0, R gives \(1 \times 1! = 1\). Total \(55\), so the rank is \(55 + 1 = 56\).
2. How many arrangements of SUCCESS are there in total?
\(\dfrac{7!}{2!\,3!} = 420\)
3. Which word comes last in the dictionary of MOTHER's letters?
TROMHE, the letters in reverse alphabetical order. Its rank is \(6! = 720\).
4. What is the rank of BOOK, and why is it so small?
3. B is the smallest letter, so nothing comes before at position 1. At position 2, K is smaller than O, which gives BKOO. At position 3, K is smaller than O again, which gives BOKO. That's 2 words before, so the rank is 3.
Exam tips for JEE preparation
1. Always write the sorted letters first. Most wrong answers come from a wrong alphabetical order.
2. Cross out letters as you use them, so you never count a used letter as "smaller".
3. Don't forget the +1 at the end. The sum counts the words before yours.
4. With repeats, divide every count, not just the total. This is the #1 trap in JEE Advanced versions of the question.
Frequently asked questions
Does the word need to have a meaning?
No. Every arrangement counts, even if it isn't a real English word.
Is the rank the same as the number of words before it?
No, it's one more. If 308 words come before MOTHER, MOTHER is the 309th word.
Which chapter is this in?
Permutations and Combinations, NCERT Class 11 Maths. It also appears in JEE Main previous year questions.
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