Dictionary Order & Alphabet
🔒 Log in to trackDictionary rank and word surgery
🔒 Log in to trackTwo specialist questions:
- Rank of a word (e.g. the dictionary rank of MACHINE among arrangements of its letters): step letter by letter. At each step count how many unused letters are smaller than the current letter — multiply that count by (remaining letters − 1)! and add. Add 1 at the end. This is 'how many words start with earlier letters' accumulation.
- Letter surgery: rearranging a word's letters alphabetically, counting letters that keep their place, or rebuilding words from selected positions — always write the derived sequence out.
The formula as written assumes all letters are distinct (every SSC rank question so far uses distinct letters). With repeated letters each block count must also be divided by the factorials of the repeats among the remaining letters.
Detailed notes
Rank of a word, and other letter-bag questions
Two families live here. Both are procedures — learn the steps once and every instance falls.
Family 1 — 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 counts how many arrangements come before the word, plus one. Procedure:
- Write the word's letters in alphabetical order (E, I, M, P, R for PRIME).
- Scan the word left to right. At each position, count how many unused letters are smaller than the current letter. Each smaller letter would start a block of arrangements: add (count) × (arrangements of the remaining letters).
- 'Arrangements of the remaining letters' = factorial of how many letters are left — divided by the factorials of any repeated-letter counts (a word with two E's has half as many distinct arrangements).
- Fix the current letter as used and move right. The final total + 1 is the rank.
Example skeleton for PRIME: P is 4th smallest of {E,I,M,P,R} → 3 × 4! before it; then R is 4th of {E,I,M,R} → 3 × 3!; then I is 2nd of {E,I,M} → 1 × 2!; then M is 2nd of {E,M} → 1 × 1!; E is 1st → 0. Rank = 72 + 18 + 2 + 1 + 1 = 94.
Family 2 — alphabetised spellings. "If the letters of TABLE are rearranged in alphabetical order, how many letters keep their position?" Write the word, write its letters sorted (A B E L T), and compare column by column. Only exact matches count. TABLE: only L (position 4) matches → one letter.
Family 3 — form the word from the letters. "Which word can be formed from the letters of CONVERSATION?" Check each option letter by letter against the bag: cross off a bag letter for every use. Two details decide these: repeats (the bag has two O's and two N's — using three O's fails) and missing letters (one absent letter kills the option). Scan for the rarest letter in each option first; that fails or passes fastest.
Family 4 — neighbours in the arrangement list. "All arrangements of BAT in dictionary order — which is 4th?" The list is short: write it out (ABT, ATB, BAT, BTA, TAB, TBA) and count. For 3 distinct letters the whole list is 6 words; writing it beats any formula.
Quick revision
- Rank = count smaller-letter blocks position by position + 1; divide by repeat factorials.
- Alphabetise-and-compare: only exact same-position letters count.
- Word-formation: cross off bag letters; watch double letters and one missing letter.
- 3-letter words: write all 6 arrangements and count slots.
- Sanity: the rank must not exceed n! (7! = 5040 for a 7-letter word).
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Dictionary rank of a wordvery common3 practice Q
'All arrangements listed as in a dictionary — what is the rank of the word?'
- Sort the letters alphabetically; scan the word left to right.
- At each position add (unused letters smaller than the current one) × (arrangements of the rest).
- Arrangements of the rest = factorial of the count, divided by repeat factorials.
- Fix the letter, move right; total + 1 = rank.
Why it works: each smaller unused letter heads a full block of arrangements that come first.
Example: Find the dictionary rank of the word CAB.
C: A and B smaller → 2 × 2! = 4. A fixed; B: nothing smaller. Rank = 4 + 1 = 5 (ABT-style list: ABC, ACB, BAC, BCA, CAB).
Type 2: Same-position count after alphabetisingcommon2 practice Q
'Letters rearranged in alphabetical order — how many letters occupy their original position?'
- Write the word; write its letters sorted beneath it.
- Compare column by column; count exact matches only.
- Answer as a number word if the options say so.
Why it works: a position survives only if the sorted letter equals the original letter there.
Example: If the letters of DERAIL are rearranged in alphabetical order, how many letters remain in their original position?
DERAIL vs ADEILR: D≠A, E≠D, R≠E, A≠I, I≠L, L≠R → None.
Type 3: Forming words from a letter bagcommon2 practice Q
'Which word can/cannot be formed from the letters of X?'
- List the bag with repeat counts.
- For each option, check its rarest letter first; cross off bag letters as used.
- One missing letter, or one letter needed more often than it appears, kills the option.
Why it works: forming a word is a multiset check — counts matter as much as presence.
Example: Which word CANNOT be formed from the letters of DISTRIBUTION?
Bag has D,I,S,T,R,B,U,O,N (I×2, T×2). STUDIO ✓, BIRD ✓, ROBUST ✓, BRUSH needs H → cannot be formed.
Type 4: Neighbours in a short arrangement listcommon2 practice Q
3-letter words: 'which is the 4th arrangement' or 'what comes just after X' in dictionary order.
- List all 3! = 6 arrangements in dictionary order (a few seconds of writing).
- 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.
Example: All arrangements of FAN are listed in dictionary order. Which word comes immediately after AFN?
List: AFN, ANF, FAN, FNA, NAF, NFA → after AFN = ANF.
Formulas
At each letter: c_i = unused letters smaller than it, r_i = letters remaining after it.
Shortcut tricks
⚡ Count smaller unused letters
Freeze the sorted letter list, and for each letter of the word (left to right) ask: how many still-unused letters are SMALLER? Multiply by the factorial of what remains, sum, +1.
Example: Q. Rank of CAB among ABC arrangements?
Sol. C: A,B smaller → 2×2! = 4. A: none smaller → 0. B: none → 0. Rank = 4 + 0 + 0 + 1 = 5.
Per-letter contribution = smaller-unused × (remaining)!.
Where students lose marks
Forgetting the final +1 (the word itself is not counted by the sums).
Using the original remaining count instead of the shrinking one.
Rearranging letters alphabetically but comparing with the original word wrongly.
Practice sets — 12 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 12 questions
Suggested time 10 min · wrong answers go to your mistake notebook automatically.