Combination:

Combination is a mathematical concept that deals with selecting a subset of objects from a larger set without considering the order of selection. It is used to calculate the number of ways in which a certain number of objects can be chosen from a larger set.

Formula:

The formula to calculate the number of combinations is given by:

C(n, r) = n! / (r! * (n-r)!)


where n is the total number of objects in the set and r is the number of objects to be selected.

Explanation:

The concept of combination is closely related to permutations. While permutations consider the order of selection, combinations do not. In other words, combinations focus on the selection of objects, regardless of their arrangement.

When calculating combinations, it is important to use the formula mentioned above. This formula accounts for the number of ways the objects can be arranged within the subset and divides it by the number of ways the objects can be arranged outside the subset.

Let's take an example to understand this better. Suppose we have a set of 5 letters: A, B, C, D, and E. We want to select 3 letters at a time. Using the combination formula, we can calculate the number of ways to do this:

C(5, 3) = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4 * 3!) / (3! * 2 * 1) = 10


Therefore, there are 10 different ways to select 3 letters from the given set.

Key Features of Combination:

- Order does not matter: Combinations only focus on the selection of objects, regardless of their arrangement or order.
- Subset selection: Combination calculates the number of ways a subset of objects can be chosen from a larger set.
- Formula-based: Combinations are calculated using the specific formula C(n, r) = n! / (r! * (n-r)!).
- Used in various fields: Combinations have applications in probability, statistics, combinatorial mathematics, and other areas.

Combination is an essential concept in mathematics and has various practical applications. Whether it is selecting a team from a group of players or choosing a committee from a pool of candidates, understanding combinations allows us to calculate the number of possible outcomes accurately.