Permutation and Combination

Example 1: Ball-drawing problem in probability. A bag contains 5 red balls and 3 white balls. Two balls are drawn at random; what is the probability of getting exactly one red and one white ball? First, the total number of possible outcomes is the number of ways to choose 2 balls out of 8 without regard to order—a combination problem calculated as C(8,2) = 28. Next, the number of favorable outcomes is the number of ways to choose 1 red ball from 5 (C(5,1) = 5) and 1 white ball from 3 (C(3,1) = 3). By the multiplication principle, there are 5 × 3 = 15 favorable cases. Therefore, the probability is 15/28.

Example 2: Line-up and seating arrangement. Five people are lining up, with the condition that person A and person B must stand together. Treat A and B as a single unit ("bonded"), resulting in 4 units to permute, giving P(4,4) = 24 arrangements. Within the bonded unit, A and B can switch places in P(2,2) = 2 ways. By the multiplication principle, the total number of valid arrangements is 24 × 2 = 48. This illustrates the application of permutations where order matters.

1. The essential difference between permutation and combination is whether order is considered.
2. Permutations emphasize order; combinations do not.
3. Widely applied in probability theory, computer science, statistics, physics, biology, and other fields.
4. When solving problems, use the addition principle (for case categorization) and multiplication principle (for step-by-step counting).
5. Master specific calculation methods including full permutations, partial permutations, combinations, permutations with repetition, and derangements.