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Combinations are a way of counting the number of unique subsets that can be formed from a set of elements. In other words, combinations answer the question “how many different ways can I choose a certain number of items from a larger set of items?”
The formula for calculating the number of combinations is:
C(n,r) = n! / (r! * (nr)!)
Where:
n is the total number of items in the set
r is the number of items being chosen
! represents the factorial operation
For example, if you have a set of 5 items (n=5) and you want to choose 3 of them (r=3), the number of possible combinations would be:
C(5,3) = 5! / (3! * (53)!)
= 120 / (6 * 2)
= 10
So there are 10 unique ways to choose 3 items from a set of 5 items.
Combinations are commonly used in probability, combinatorics, and various other areas of mathematics and computer science. They are a fundamental concept for understanding how to count the number of possible outcomes in situations involving selection or arrangement of elements.