Permutations and Combinations Overview

Permutations and Combinations Overview

• Permutation refers to the arrangement of items in a specific sequence or order.
• Combination refers to the selection of items without regard to the order.

Basic Concepts

• A person can wear either a shirt or a t-shirt. If he has 3 shirts and 4 t-shirts, he has 7 total options.

Routes and Pathways

• From city A to city B, there are 4 roads; from city B to city C, there are 5 roads.
• The total number of ways to travel from city A to city C via city B is the product of the roads from A to B and B to C, which is (4 \times 5 = 20).

Choosing Students

• In a class with 15 boys and 10 girls, a teacher can choose one boy and one girl in (15 \times 10 = 150) ways.
• The total ways to select one student for a monitor position from 25 students (15 boys and 10 girls) is 25.
• For a captain and vice-captain role, the teacher can select 2 distinct students in (C(25, 2) = \frac{25!}{2!(25-2)!} = 300) ways.

Count of Paths and Selections

• Total roads for the journey from A to C through B include:
  • 4 roads from A to B,
  • 5 roads from B to C,
  • 2 additional roads from A to C directly.
• To travel from B to C, the number of ways equals 5 (roads available).

Multiple Prize Distributions

• For distributing 1st, 2nd, and 3rd prizes among 7 students, the combinations depend on whether order matters. If it does (as in a race), that is a permutation scenario.

Chairman and Vice-Chairman Selection

• In a committee of 8 individuals, the selection of a chairman and vice-chairman (where one person cannot hold both positions) can be calculated as (P(8, 2) = 8 \times 7 = 56).

Flag Design Combinations

• To design a flag with 5 vertical strips using the colors red, yellow, or blue:
  • Each strip can be any of the 3 colors; thus, the total combinations are (3^5 = 243) different flag designs.