Combinations

Combinations are a way of selecting items from a larger set without considering the order in which they are selected. In other words, combinations are used to calculate the number of ways to choose a specific number of items from a larger set, without regard to the order in which the items are chosen.

Formula for Combinations

The formula for calculating combinations is given by:

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

• C(n, r) represents the number of combinations of n items taken r at a time
• n! represents the factorial of n (i.e., the product of all positive integers from 1 to n)
• r! represents the factorial of r
• (n-r)! represents the factorial of (n-r)

Example

Let's say you have 5 different colored balls, and you want to choose 3 of them. Using the combination formula, C(5, 3) = 5! / (3!(5-3)!) = 10. This means there are 10 different combinations of 3 balls that can be chosen from the set of 5.

Study Guide

Here are some key points to remember when working with combinations:

1. Combinations are used when the order of selection does not matter.
2. The combination formula is C(n, r) = n! / (r!(n-r)!).
3. Factorials are an important part of the combination formula, and are denoted by the symbol "!".
4. When calculating combinations, it's important to pay attention to the number of items being chosen (r) and the size of the larger set (n).
5. Combinations can be used in various real-world scenarios, such as selecting a committee from a group of people, choosing a team from a pool of players, or picking a certain number of items from a collection.

Understanding combinations is an important concept in mathematics, and it can be applied to a wide range of problems and scenarios. Practice using the combination formula and solving problems involving combinations to strengthen your understanding of this topic.