Combination of n things taking r at a time C(n,r)
Combination of n things taking r at a time C(n,r)
In the study of combinatorics, the concept of combinations is a fundamental counting principle used to find the number of ways to select items from a collection, such that the order of selection does not matter. This is denoted as C(n, r), where n represents the total number of items and r represents the number of items to be selected.
Understanding Combinations
Combinations can be understood as the different groups or subsets that can be formed from a larger set when the order of the elements is not important. For example, if we have a set of letters {A, B, C}, the combinations of 2 letters are {A, B}, {A, C}, and {B, C}.
Formula for Combinations
The number of combinations of n things taken r at a time is given by the formula:
[ C(n, r) = \frac{n!}{r!(n-r)!} ]
where n! denotes the factorial of n, which is the product of all positive integers up to n.
Table of Differences and Important Points
| Permutations (P) | Combinations (C) |
|---|---|
| Order matters | Order does not matter |
| P(n, r) = $\frac{n!}{(n-r)!}$ | C(n, r) = $\frac{n!}{r!(n-r)!}$ |
| Used when arranging items | Used when selecting items |
| More outcomes than combinations | Fewer outcomes than permutations |
Examples
Let's look at some examples to understand how to calculate combinations.
Example 1: Basic Combination
Suppose we have 5 books, and we want to select 2 to take on a trip. How many different pairs of books can we choose?
Using the combination formula:
[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10 ]
So, there are 10 different pairs of books we can choose.
Example 2: Combination with Large Numbers
Imagine a lottery where you must choose 6 numbers out of 49. How many different combinations of numbers can you select?
[ C(49, 6) = \frac{49!}{6!(49-6)!} = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6 \times 5 \times 4 \times 3 \times 2 \times 1} ]
This calculation can be simplified by canceling out common factors:
[ C(49, 6) = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 13,983,816 ]
Therefore, there are 13,983,816 different combinations of numbers that can be selected in the lottery.
Example 3: Identical Items
If we have 8 identical pens and we want to give 3 pens to a friend, how many different ways can we do this?
Since the pens are identical, there is only one way to select 3 pens out of 8.
[ C(8, 3) = 1 ]
Conclusion
The concept of combinations is a crucial part of combinatorics and is widely used in various fields such as mathematics, statistics, and computer science. Understanding how to calculate combinations allows us to solve problems related to selection and arrangement where the order is not a concern.