Permutations
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision.
Permutations — Key Facts for CUET • Core formula: $^nP_r = \dfrac{n!}{(n-r)!}$; when arranging all $n$ objects, $^nP_n = n!$ • Most tested CUET concept: Arranging $n$ objects taken $r$ at a time, especially $^nP_r$ and special cases like $^nP_0 = 1,\ ^nP_1 = n,\ ^nP_n = n!$ • Common mistake students make: Treating permutations with repetition the same as without repetition — when objects are identical, divide by factorials of frequencies: $\dfrac{n!}{p!,q!,r!\ldots}$ • Key technique: Use slot method (fill positions one by one) for arrangements with restrictions. For $r$ positions from $n$ distinct objects: multiply descending numbers → $n \times (n-1) \times \cdots \times (n-r+1) = \dfrac{n!}{(n-r)!}$ • Important exception: Circular permutations — for $n$ distinct objects around a circle: $(n-1)!$. Direction matters (clockwise ≠ anticlockwise). • Most frequent question type: “In how many ways can $k$ objects be arranged from $n$?” (direct formula application) and “arrange the letters of the word ’$\ldots$’ ” (identical object division) ⚡ Exam tip: In “arrange the letters” questions, ALWAYS check for repeated letters FIRST — divide $n!$ by factorials of each repeated letter’s count before applying any other condition.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Permutations — CUET Study Guide
Permutation means an ordered arrangement of distinct objects. If you choose and arrange $r$ objects out of $n$ distinct objects, the number of ways is $^nP_r = \dfrac{n!}{(n-r)!}$. This formula follows directly from the Fundamental Principle of Counting — multiply choices for each position: $n \times (n-1) \times \cdots \times (n-r+1)$.
Key formulas to memorize: $^nP_r = \dfrac{n!}{(n-r)!}$; $^nP_n = n!$; $^nP_0 = 1$. For circular arrangements, rotate to eliminate identical rotations: $(n-1)!$.
CUET question patterns: The most common pattern is arranging letters of a word (with or without repetition), arranging people in a row or around a table with restrictions (e.g., “A and B must sit together”). For “must sit together” problems, treat the grouped pair as one entity, then multiply arrangements of the group by internal arrangements ($2!$ for two people).
Common traps:
- Identical letters require division by factorials of frequencies.
- “Selection” wording implies combination, not permutation — watch for it.
- In circular permutations, $(n-1)!$ only applies when reflections are considered different (i.e., no clockwise/anticlockwise symmetry).
Solved Example 1: Arrange the letters of “DELHI”. $\rightarrow 5! = 120$ (all distinct). Solved Example 2: Arrange the letters of “BANANA”. $\rightarrow \dfrac{6!}{3! \times 2!} = \dfrac{720}{12} = 60$ (A repeats 3 times, N repeats 2 times).
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer timeline.
Permutations — Comprehensive CUET Notes Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.