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### Combinations (simple)

Definition

Combinations are the groups of objects taken from a larger group of objects. When you are given a large group of all different objects, and you select some of the objects randomly, then the selected group is known as a combination.

Explanation
 The topic combinations in math is of profound importance in applied mathematics and is also used to in the derivation of the Binomial Theorem. In combinations, you have to form groups of a specific number of objects by selecting random objects from a large group of different objects. Each of the groups has to contain at least one different object than the rest. Any two or more groups that contain the same set of objects is not considered valid. Each group is called a combination. Here we will learn how to calculate the number of combinations when a specific number of objects are selected at random from a large group of different objects.
For example, there are 10 objects all off different color. Now you have to from combinations of 5 objects from these 10 objects. How many different combinations of 5 objects will you be able to form when you can choose from 10 different objects?

This can be determined by the use of the following formula:

Formula for Combinations:

 The number of combinations of 'n' objects taken 'r' at a time is given by Where, n = Total number of objects. r = Number of objects taken in each combination. The symbol "!" in the above formula means factorials. To learn about factorials, go here: Introduction to factorial notation.

Applying the above formula: (Solved Example)
Let us learn the above formula with the help of an exmaple. Solving the following question with the help of the above formula:
Question: Find the number of combinations from a group of 10 different objects if 5 objects are taken in each combination.
 Solution: