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NDA Maths · Permutation & Combination

Permutations & Restricted Arrangements

Arranging objects in order — distinct or with repeated letters — and handling the standard restrictions: things together, things apart, and fixed positions.

All Permutation & Combination notesNDA Maths strategy

Why this matters

Word-arrangement questions are a guaranteed NDA appearance. A few reliable moves — divide by repeats, treat a group as one block, fill restricted slots first — cover almost all of them.

Concept 1 of 2

Arranging objects (with repeats)

Intuition

Arranging nn distinct objects gives n!n! orders. When letters repeat, identical arrangements are over-counted, so divide by the factorial of each repeat count.

Definition

nn distinct objects: n!n! arrangements. With repeats — nn objects where one letter appears pp times, another qq times, etc.: n!p! q!⋯\dfrac{n!}{p!\,q!\cdots}. (E.g. MATHEMATICS: 11 letters with M, A, T each twice ⇒ 11!2! 2! 2!\dfrac{11!}{2!\,2!\,2!}.)

Worked example

How many arrangements of the letters of LEVEL are there?
  1. 5 letters with L twice and E twice.
  2. 5!2! 2!=1204=30\dfrac{5!}{2!\,2!}=\dfrac{120}{4}=30.
Answer:3030.
Practice this conceptself-check · 4 quick reps

Try it yourself

How many distinct words can be formed from the letters of DELHI?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    Arrangements of nn distinct objects?
  2. 2.
    Arrangements of MATHEMATICS?
  3. 3.
    Why divide by repeat-factorials?
  4. 4.
    Arrangements of LEVEL?

From the bank · past-year question

Example 1Permutation & CombinationEASY
What is the number of ways in which the letters of the word 'ABLE' can be arranged so that the vowels occupy even places?

[Q14 · Apr · 2020]

Drill 4 more on arranging objects (with repeats)

Concept 2 of 2

Restrictions: together, apart, fixed positions

Intuition

Restrictions reshape the count. 'Together' → glue the group into one block. 'Apart/alternating' → place the others first, then drop the rest into the gaps. 'Fixed slots' (even positions, vowels in odd places) → fill the constrained slots first, then the rest.

Definition

  • Together (block): treat the kk items as one unit ⇒ (n−k+1)!(n-k+1)! for the units × k!\times\,k! inside.
  • Apart / alternating: arrange the unrestricted items, then choose gaps for the rest.
  • Fixed positions: fill the restricted positions first (e.g. vowels into the even slots), then fill the remaining positions.

Worked example

In how many ways can 4 boys and 3 girls sit in a row with all girls together?
  1. Glue the 3 girls into one block ⇒ 5 units (4 boys + block): 5!5!.
  2. Girls within the block: 3!3!. Total =5!×3!=720=5!\times 3!=720.
Answer:720720.
Practice this conceptself-check · 4 quick reps

Try it yourself

How many arrangements of TIGER have both vowels in the two even positions?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    'Together' is handled by which method?
  2. 2.
    Block of kk among nn: factor for inside?
  3. 3.
    'Apart' is handled by?
  4. 4.
    Fixed-position items: fill which slots first?

From the bank · past-year question

Example 2Permutation & CombinationMODERATE
Consider the following for the items that follow: Consider the word 'QUESTION':
How many 8-letter words with or without meaning, can be formed such that consonants and vowels occupy alternate positions?

[Q26 · Sep · 2022]

Drill 11 more on restrictions: together, apart, fixed positions

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Permutation & CombinationHARD
If different permutations of the letters of the word 'MATHEMATICS' are listed as in a dictionary, how many words (with or without meaning) are there in the list before the first word that starts with C?

[Q3 · Sep · 2022]

Example 2Permutation & CombinationMODERATE
How many permutations are there of the letters of the word 'TIGER' in which the vowels should not occupy the even positions?

[Q16 · Apr · 2022]

Example 3Permutation & CombinationEASY
In how many ways can the letters of the word DELHI be arranged keeping the positions of vowels and consonants unchanged?

[Q18 · Apr · 2025]

Example 4Permutation & CombinationHARD
The number of different words (eight-letter words) ending and beginning with a consonant which can be made out of the letters of the word 'EQUATION' is

[Q19 · Apr · 2017]

Example 5Permutation & CombinationEASY
How many different permutations can be made out of the letters of the word 'PERMUTATION'?

[Q10 · Sep · 2017]

Drill every past-year question on this subtopic

17 questions from the bank — paginated, with cart and Word-export support.

Drill the 17 questions

Related notes