CIE A Level Maths: Probability & Statistics 1

2.2 Permutations & Combinations

Contents

• 2.2.1 Arrangements & Factorials
• 2.2.2 Permutations
• 2.2.3 Combinations

2.2.1 Arrangements & Factorials

Arrangements

• Arranging Different Objects:
  • Number of ways to arrange (n) different objects is (n!) (factorial).
  • For example, arranging A, B, C:
    • Options: ABC, ACB, BAC, BCA, CAB, CBA.
• Identical Objects:
  • If objects are identical, divide the total arrangements by the factorial of identical objects.
  • Example: Arranging A, A, C:
    • Total: 6 ways, reduce duplicates: (\frac{6}{2!} = 3) ways.

Factorials

• Definition:
  • Factorial (n! = n \times (n-1) \times (n-2) \cdots \times 1).
  • Special case: (0! = 1).
• Properties:
  • (n! = n \times (n-1)!).
  • Useful for cancelling terms in permutations and combinations.

2.2.2 Permutations

Definition

• Permutations involve arranging (n) items where the order matters.
  • Formula: (n!) for (n) items.
• Condition with Repeated Items:
  • If an item repeats (r) times in (n) items, divide by (r!).

Calculating Permutations

• Finding (r) Permutations of (n):
  • Example: Arrange 3 out of 5 items: (5 \times 4 \times 3) or (\frac{5!}{2!}).
• Handling Restrictions:
  • Items together: Treat as single unit, multiply internal arrangements.
  • Items separated: Subtract arrangements where items are together from total.

2.2.3 Combinations

Definition

• Combinations: Arrangements where order doesn't matter.
  • Formula: (\binom{n}{r} = \frac{n!}{(n-r)!r!}).

Calculating Combinations

• Examples:
  • Choose 2 from 3 items (A, B, C):
    • Total permutations: 6, divide by 2 (order): 3 combinations.
  • Choose 3 from 5 items (A, B, C, D, E):
    • Total permutations: 60, divide by 6 (order): 10 combinations.

Handling Identical Objects

• Consider identical items separately in calculations.

Key Points

• Multiplication vs Addition:
  • Multiply when conditions require 'and'.
  • Add when conditions allow 'or'.
• Probability:
  • Probability questions can use combinations for calculating outcomes.

Examiner Tips

• Pay Attention to Wording:
  • Keywords like "arrange" vs "choose" indicate permutations vs combinations.
  • Conditions about items being together or separated significantly change the solution approach.