In mathematics, combination refers to the selection of items from a larger set without considering the order of selection. It is a fundamental concept in combinatorics, which deals with counting and arranging objects.
The concept of combination has been studied for centuries. The ancient Greeks and Chinese mathematicians explored various aspects of combinations in their works. However, the formal development of combination theory can be attributed to the French mathematician Blaise Pascal in the 17th century.
The concept of combination is typically introduced in middle or high school mathematics, depending on the curriculum. It is commonly covered in courses such as algebra, probability, and combinatorics.
Combination involves several key knowledge points, including:
Selection without order: Combination focuses on selecting items without considering their order. For example, choosing three fruits from a basket without caring about the order in which they are picked.
Binomial coefficients: Combination utilizes binomial coefficients, which represent the number of ways to choose a certain number of items from a larger set. These coefficients are often denoted as "n choose k" or written as C(n, k).
Permutations vs. combinations: It is important to distinguish between permutations and combinations. Permutations consider the order of selection, while combinations do not. For example, selecting three different books from a shelf would be a combination, while selecting three books in a specific order would be a permutation.
There are two main types of combinations:
Combination with repetition: This type allows for the selection of items with replacement. For example, choosing three candies from a jar where duplicates are allowed.
Combination without repetition: This type does not allow for the selection of items with replacement. For example, selecting three students from a class to form a committee, where each student can only be chosen once.
Some important properties of combination include:
Symmetry: The number of ways to choose k items from a set of n items is equal to the number of ways to choose n-k items. In other words, C(n, k) = C(n, n-k).
Pascal's identity: Pascal's identity states that the sum of two adjacent binomial coefficients is equal to the binomial coefficient of the next row. This can be expressed as C(n, k) + C(n, k+1) = C(n+1, k+1).
To calculate the number of combinations, the formula or equation for combination is used. The formula is:
C(n, k) = n! / (k! * (n-k)!)
Where n is the total number of items and k is the number of items to be chosen.
To apply the combination formula, substitute the values of n and k into the equation and perform the necessary calculations. The result will give you the number of combinations possible.
The symbol commonly used to represent combination is "C". It is often written as "n choose k" or expressed as C(n, k).
There are several methods for solving combination problems, including:
Example 1: In a deck of cards, how many ways can you choose 5 cards? Solution: Using the combination formula, C(52, 5) = 52! / (5! * (52-5)!) = 2,598,960 ways.
Example 2: A pizza place offers 10 different toppings. How many different combinations of 3 toppings can you choose? Solution: C(10, 3) = 10! / (3! * (10-3)!) = 120 ways.
Example 3: A committee of 4 students needs to be formed from a class of 20 students. How many different combinations of the committee can be chosen? Solution: C(20, 4) = 20! / (4! * (20-4)!) = 4,845 ways.
Q: What is combination? Combination refers to the selection of items from a larger set without considering the order of selection.
Remember, combination is a fascinating concept that finds applications in various fields, including probability, statistics, and computer science. Understanding the fundamentals of combination is essential for tackling more advanced mathematical problems.