Permutation and combination are two important concepts in combinatorics, the branch of mathematics that deals with counting and arranging objects. They are often used in probability and statistics, as well as in many other areas of mathematics and science.

Permutation: A permutation is an arrangement of objects in a specific order. For example, if we have three letters (A, B, and C), there are 3! (3 factorial) possible permutations, which is 6. The permutations are: ABC, ACB, BAC, BCA, CAB, CBA.

Combination: A combination is a selection of objects without regard to the order in which they are arranged. For example, if we have three letters (A, B, and C) and want to select two of them, there are 3C2 (3 choose 2) possible combinations, which is 3. The combinations are: AB, AC, BC.

Example 1: A teacher wants to form a committee of 4 students out of a class of 10. How many different committees can be formed? This is a permutation problem because the order of the students on the committee matters. The answer is 10P4 = 5040.

Example 2: A store owner wants to select a team of 3 employees out of a pool of 10 to work on a special project. How many different teams can be formed? This is a combination problem because the order of the employees on the team doesn’t matter. The answer is 10C3 = 120.

Example 3: A sports team has 6 players and wants to select a lineup of 4 players for the next game. How many different lineups are possible? This is a permutation problem because the order of the players on the lineup matters. The answer is 6P4 = 360.

Example 4: A company has 12 employees and wants to select a group of 4 employees to attend a training seminar. How many different groups can be formed? This is a combination problem because the order of the employees in the group doesn’t matter. The answer is 12C4 = 495.

In conclusion, permutation and combination are important concepts in combinatorics that are used to count and arrange objects. They are used in many areas of mathematics and science, such as probability and statistics. It’s important to understand when to use permutations and when to use combinations, as they give different results depending on the problem at hand.