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Dictionary Order & Alphabet

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medium importance~1-2 Q in Tier 13 formulas⚡ 6 shortcuts4 subtopics
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Dictionary rank and word surgery

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⏱ 5 min read🧩 5 question types🎯 12 practice Q
The idea in one minute

This subtopic has two specialist questions. First, the dictionary rank of a word among all arrangements of its letters: count how many arrangements start with a smaller letter, block by block, then add one. Second, letter surgery: alphabetise a word and count letters that stay put, or test which words can be built from a word's letters. Both are written procedures. Do them on paper, step by step.

01

Two families of questions

The first family asks for the dictionary rank of a word among all arrangements of its letters. The second does letter surgery: alphabetise a word, or build words from its letters. Both are fixed procedures.

02

The dictionary rank of a word

"The letters of PRIME are arranged in all possible ways, as in a dictionary. What is the rank of PRIME?" The rank is one more than the number of arrangements that come before PRIME.

Write the letters in order first: E, I, M, P, R. Now walk the word:

  1. First letter P. Unused letters smaller than P: E, I, M → 3. Each heads a block of 4! = 24 words. Add 3 × 24 = 72.
  2. Fix P. Next letter R. Unused smaller: E, I, M → 3 × 3! = 18.
  3. Fix R. Next letter I. Unused smaller: E → 1 × 2! = 2.
  4. Fix I. Next letter M. Unused smaller: E → 1 × 1! = 1.
  5. Fix M. Last letter E. Nothing smaller → 0.

Rank = 72 + 18 + 2 + 1 + 0 + 1 = 94.

Rule: At each letter: (smaller unused letters) × (factorial of the letters left after it). Add up, then add 1 for the word itself.

03

Repeated letters

SSC rank questions so far use words with all-distinct letters. If a letter repeats, divide each block by the factorials of the repeats among the remaining letters. A word with two E's has half as many distinct arrangements.

Watch: Forgetting the final +1 is the most common slip. The sum counts only the words BEFORE yours.

04

Alphabetise and compare

"If the letters of TABLE are rearranged in alphabetical order, how many letters keep their position?"

Write the word and, under it, its sorted letters. Tick exact column matches only. TABLE gives A, B, E, L, T under T, A, B, L, E. Only L (slot 4) matches. Answer: one letter.

DERAIL gives A, D, E, I, L, R under D, E, R, A, I, L. No column matches. Answer: none.

Tip: Write both rows fully. Counting matches in your head misses the shifted columns.

05

Form a word from a letter bag

DISTRIBUTION gives the bag D, I×3, S, T×2, R, B, U, O, N. Question: which of Studio, Bird, Robust, Brush cannot be formed?

Brush needs H. The bag has no H, so Brush fails. The other three use only letters present, within their counts.

Steps for each option: check its rarest letter first. One missing letter kills the option. A letter used more often than it appears also kills it.

Watch: Counts matter as much as presence. A bag with one O cannot build a word with two O's.

06

Short arrangement lists

For three letters the whole list has 3! = 6 words. Write them in order and count. BAT gives ABT, ATB, BAT, BTA, TAB, TBA. Any lookup ("which is 4th", "what follows AFN") is then plain reading.

Example: All arrangements of FAN: AFN, ANF, FAN, FNA, NAF, NFA. The word after AFN is ANF.

07

From the end of the list

CAB has rank 5 among 6 arrangements. From the end it sits at 6 − 5 + 1 = 2nd. Use: total − rank + 1.

08

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Dictionary rank of a word

How to spot it:

'All arrangements listed as in a dictionary — what is the rank of the word?'

rank=1+∑ici×(ri)!\text{rank} = 1 + \sum_i c_i \times (r_i)!
Method

  1. Sort the letters; scan the word left to right.

  2. At each position add (unused letters smaller than the current one) × (factorial of letters left).

  3. Fix the letter, move right, repeat.

  4. Add 1 at the end.

Why it works:

Each smaller unused letter heads a whole block of arrangements that come before your word.

Try this

Find the dictionary rank of the word PRIME.

Show solution

  1. Sorted letters: E, I, M, P, R.

  2. P: 3 smaller → 3 × 4! = 72. R: 3 smaller → 3 × 3! = 18. I: 1 → 2. M: 1 → 1. E: 0.

  3. 72 + 18 + 2 + 1 + 0 + 1 = 94.

Answer

94

Type 2common2 practice Q

Same-position count after alphabetising

How to spot it:

'Letters rearranged in alphabetical order — how many keep their original position?'

Method

  1. Write the word; write its sorted letters beneath it.

  2. Compare column by column.

  3. Count exact matches only.

Why it works:

A letter keeps its position only if the sorted letter at that slot equals the original.

Try this

If the letters of DERAIL are rearranged in alphabetical order, how many letters stay in place?

Show solution

  1. DERAIL sorted is A, D, E, I, L, R.

  2. Compare: D-A, E-D, R-E, A-I, I-L, L-R. No column matches.

  3. Zero letters stay put.

Answer

None

Type 3common2 practice Q

Forming words from a letter bag

How to spot it:

'Which word can or cannot be formed from the letters of X?'

Method

  1. List the bag with repeat counts.

  2. For each option, check its rarest letter first; cross off bag letters as used.

  3. One missing letter, or one over-used letter, kills the option.

Why it works:

Forming a word is a count check, not just a presence check.

Try this

Which word CANNOT be formed from the letters of DISTRIBUTION: Studio, Bird, Robust, Brush?

Show solution

  1. The bag has D, I×3, S, T×2, R, B, U, O, N.

  2. Studio, Bird and Robust use only letters inside the counts.

  3. Brush needs H, which the bag does not have.

Answer

Brush

Type 4common2 practice Q

Neighbours in a short arrangement list

How to spot it:

Three-letter words: 'which is the 4th arrangement' or 'what comes just after X'.

Method

  1. List all 3! = 6 arrangements in dictionary order.

  2. Count to the asked slot, or read the requested neighbour.

Why it works:

For six items, writing the list is faster and safer than any formula.

Try this

All arrangements of FAN are listed in dictionary order. Which word comes immediately after AFN?

Show solution

  1. The list: AFN, ANF, FAN, FNA, NAF, NFA.

  2. AFN is 1st, so the next word is ANF.

Answer

ANF

Type 5occasional

Rank from the end of the list

How to spot it:

'What is the position of the word from the end / how many words come after it?'

from end=n!−rank+1\text{from end} = n! - \text{rank} + 1
Method

  1. Find the rank as usual.

  2. Total arrangements = n! (divide by repeat factorials if letters repeat).

  3. From the end = total − rank + 1.

Why it works:

The list is symmetric: rank r of n! items sits n! − r + 1 from the back.

Try this

All arrangements of CAB are listed as in a dictionary. What is CAB's position from the end?

Show solution

  1. CAB's rank is 5 (list: ABC, ACB, BAC, BCA, CAB).

  2. Total arrangements = 3! = 6.

  3. 6 − 5 + 1 = 2.

Answer

2nd

09

Formula sheet

Rank of a word
rank=1+∑ici×(ri)!\text{rank} = 1 + \sum_i c_i \times (r_i)!

c_i = unused letters smaller than the letter at position i; r_i = letters remaining after position i.

10

Shortcuts that save time

⚡ Count smaller unused letters

Freeze the sorted letter list. For each letter of the word, left to right, count how many still-unused letters are smaller. Multiply by the factorial of what remains, sum, add one.

Example

Find the dictionary rank of CAB.

Show solution

  1. Sorted letters: A, B, C. First letter C: A and B are smaller → 2 × 2! = 4.

  2. Fix C. Next letter A: nothing smaller → 0. Last letter B: nothing smaller → 0.

  3. Rank = 4 + 0 + 0 + 1 = 5.

Answer

5

⚡ Three letters: write all six

For a three-letter word the whole list has just 6 entries. Writing them beats any formula and never miscounts.

Example

All arrangements of BAT are listed in dictionary order. Which is the 4th word?

Show solution

  1. Write the six: ABT, ATB, BAT, BTA, TAB, TBA.

  2. Count to the 4th.

Answer

BTA

11

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Forgetting the final +1 in a rank sum.

The sum counts words BEFORE yours; add 1 for the word itself.

Mistake 02

Using the original remaining count instead of the shrinking one.

After fixing each letter, recount the unused letters and the positions left.

Mistake 03

Counting used letters as 'smaller unused'.

Cross out each fixed letter; only the unused ones count.

Mistake 04

In letter-bag questions, checking only presence, not counts.

A repeated letter in the option needs the same repeat in the bag.

Mistake 05

Comparing alphabetised columns by memory.

Write the word and its sorted form one under the other; tick exact matches.

12

Quick revision

Read this the night before the exam.

  • Rank = sum of (smaller unused) × (remaining factorial), plus 1.

  • Repeated letters: divide blocks by the repeat factorials.

  • Alphabetise and compare: only exact same-slot matches count.

  • Letter bag: rarest letter first; missing or over-used letters kill an option.

  • Three-letter lists: write all 6 arrangements and read.

  • From the end of the list: total − rank + 1.

13

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 10 min · wrong answers go to your mistake notebook automatically.

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