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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?
Practice this conceptself-check · 4 quick reps

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]

Repeated letters → divide by the repeat-factorials

The arrangements of a word with repeated letters is not n!n!. For each letter repeated pp times, swapping those identical copies gives the *same* word, so you over-count by p!p!. Divide: MATHEMATICS (M, A, T each twice) has 11!2! 2! 2!\dfrac{11!}{2!\,2!\,2!}, not 11!11!.
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?
Practice this conceptself-check · 4 quick reps

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

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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

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