Combination Calculation
Combinations, also known as arrangements, are an important tool in mathematics, particularly in the field of probability. They are frequently used to calculate the number of possible ways that a group of objects can be chosen from a larger set, without regard to order.
The formula for calculating the number of combinations of n objects taken r at a time is given by:
C(n,r)=r!(n−r)!n!
where "!" denotes the factorial function. Factorial is defined as the product of all positive integers up to and including the given integer. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
To understand the formula better, let's consider an example. Suppose we have 6 people (A, B, C, D, E, F) and need to choose 3 of them to form a committee. If we list all possible groups of 3 people, we will end up with 20 different combinations.
We can also use the combination formula to compute this result directly:
C(6,3)=3!(6−3)!6!
C(6,3)=3⋅2⋅16⋅5⋅4
C(6,3)=20
Therefore, there are 20 possible ways to choose a committee of 3 people from a set of 6.
It is important to note that order does not matter in combinations. For example, choosing A, B, C is the same combination as choosing C, B, A. This is why we divide by the factorials of both r and (n-r). The numerator represents the number of possible orderings of the r chosen objects, while the denominator eliminates duplicates by grouping the orderings of the same set of objects together.
In conclusion, combination calculation is an important tool for probability and has a variety of applications in other fields of mathematics like algebra and geometry. The formula C(n,r)=r!(n−r)!n! can be used to compute the number of combinations of n objects taken r at a time.