Combinations and Permutations

Factorials

• Definition: A factorial is the product of all positive integers less than or equal to an integer n.

• Notation: Denoted as n! (n factorial), pronounced as "n factorial".

• Calculation: n! = n × (n - 1) × (n - 2) × ... × 1.

  • Example: 5! = 5 × 4 × 3 × 2 × 1 = 120.

Combinations

• Definition: A combination involves choosing a specific number of objects from a group where the order does not matter.

• Notation: Denoted as C(n, r) or nCr, where:

  • n = total number of objects in the group.

  • r = number of objects to choose.

• Formula: C(n, r) = n! / (r! × (n - r)!).

  • Example: 10 choose 5 = 10C5 = 10! / (5! × (10 - 5)!) = 252.

• Application Example: Choosing 3 graduate students from a class of 20 for a trip:

  • C(20, 3) = 20! / (3! × 17!) = 1,140 combinations.

Permutations

• Definition: A permutation involves choosing a specific number of objects from a group where the order does matter.

• Notation: Denoted as P(n, r) or nPr, where:

  • n = total number of objects in the group.

  • r = number of objects to choose.

• Formula: P(n, r) = n! / (n - r)!.

  • Example: 10 permute 5 = 10P5 = 10! / (10 - 5)! = 30,240 different ways when order matters.

• Application Example: Selecting 3 students where 1st gets a full trip, 2nd pays half, and 3rd pays full:

  • P(20, 3) = 20! / (20 - 3)! = 6,840 ways.