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compulsory · CSS study notes

19. Permutations

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 19 (Permutations)

Past Paper Weightage

  • Moderate-high in MPT/General Ability papers — factorial-based arrangement questions in small groups are a recurring, fast-scoring question type once the formula is automatic.

Key Concepts and Formula

  • Factorial (n!): the product of all positive integers up to n
n!=n×(n−1)×(n−2)×⋯×1n! = n\times(n-1)\times(n-2)\times \dots \times 1

by definition, 0! = 1

  • Permutation: an arrangement of items where order matters ("ranking," "arranging," "forming a sequence/password/queue")
  • Formula: number of ways to arrange r items chosen from n distinct items:
nPr=n!(n−r)!{}^nP_r = \frac{n!}{(n-r)!}
  • Special case: arranging ALL n items (r=n) simplifies to just n!, since (n−n)! = 0! = 1
  • Repetition/identical items: if some items in a full set of n are identical (e.g., letters repeating in a word), divide n! by the factorial of the count of each repeated item, to avoid overcounting indistinguishable arrangements:
Distinct arrangements=n!p! q! …\text{Distinct arrangements} = \frac{n!}{p! \, q! \, \dots}

where p, q,... are the counts of each repeated item

Worked Example 1 (Arranging All Items)

  • In how many ways can 5 different books be arranged on a shelf?
    • Step 1: Since all 5 books are being arranged (r=n=5), use the special case: total arrangements = 5!.
    • Step 2: Calculate 5! = 5×4×3×2×1.
    • Step 3: 5×4=20, 20×3=60, 60×2=120, 120×1=120 → 120 ways.

Worked Example 2 (Choosing and Arranging a Subset)

  • In how many ways can a 3-digit code be formed using 5 distinct digits (1,2,3,4,5), with no digit repeated?
    • Step 1: This is a permutation of r=3 items chosen from n=5, since order matters (a code) and no repeats are allowed: ⁵P₃ = 5!/(5−3)!.
    • Step 2: Calculate: 5!/2! = (5×4×3×2×1)/(2×1) = 120/2 = 60.
    • Step 3: Answer: 60 codes.
    • Shortcut check: you can also think of it directly — 5 choices for the first digit, 4 remaining choices for the second, 3 remaining for the third: 5×4×3 = 60, matching the formula result.

Worked Example 3 (Repeated Letters)

  • In how many distinct ways can the letters of the word "LEVEL" be arranged?
    • Step 1: Count total letters: L-E-V-E-L has 5 letters, with "L" repeated twice and "E" repeated twice.
    • Step 2: Start with the total arrangements if all letters were distinct: 5! = 120.
    • Step 3: Divide by the factorial of each repeated letter's count to remove duplicate/indistinguishable arrangements: 120 ÷ (2! × 2!) = 120 ÷ (2×2) = 120÷4 = 30 distinct arrangements.

Worked Example 4 (Small Group — Seating)

  • In how many ways can 4 people be seated in a row of 4 chairs?
    • Step 1: All 4 people fill all 4 chairs, so r=n=4 — use the special case: 4!.
    • Step 2: Calculate: 4×3×2×1 = 24.
    • Step 3: Answer: 24 ways.

MCQ Traps

  • Permutation (order matters) vs Combination (order doesn't matter, see the Probability & Counting page) — the single biggest source of errors in this topic; the keyword test is: "arrange/rank/order/code/password" → permutation, "choose/select/form a group/committee" → combination
  • Forgetting to divide by repeated-item factorials when a word/set has duplicate elements is a very common miscalculation

Self-Test Questions

  1. In how many ways can 6 different trophies be arranged on a shelf? Show the factorial calculation.
  2. How many 4-letter codes can be formed from 7 distinct letters, with no letter repeated? Show both the formula method and the direct-multiplication shortcut.
  3. In how many distinct ways can the letters of the word "STATISTICS" be arranged? (Hint: count each repeated letter carefully before dividing.)

Past-Paper MCQs (FPSC)

✅

Taken from the FPSC papers you shared (2015–2025). FPSC does not publish MCQ keys, so the answer is worked out by me.

  1. (CSS 2024, Q1.3) What are total number of combinations of 09 things taken 03 at a time? (A) 18 (B) 27 (C) 48 (D) 84
  • ✅ Answer key
    1. (D) 84, since ⁹C₃ = 9! ÷ (3! × 6!) = (9 × 8 × 7) ÷ 6 = 84

Past-Paper Questions with Answers (FPSC)

✅

No descriptive question in the papers you shared asked directly for a permutation or combination calculation; ordering problems appear as arrangement puzzles.

  • Related past questions on ordering and arrangement
    • CSS 2019 Q8(a): ordering seven piano students under restrictions (answer: Z plays seventh); see the Arrangement and Seating Puzzles chapter
    • CSS 2020 Q7(c): seating nine people in three cars (answer: Car-1 A, D, W; Car-2 Z, E, X; Car-3 B, C, Y)
    • Formulas to remember: ⁿPᵣ = n! ÷ (n − r)! (order matters); ⁿCᵣ = n! ÷ (r!(n − r)!) (order does not matter)

CSS Exam Perspective

🎯

  • How it is tested: one MCQ on combinations (2024) and constraint puzzles
  • Tip: ask 'does order matter?': yes means permutation, no means combination