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Introduction

Combination is a fundamental concept in mathematics that deals with selecting a set of objects from a larger group, without regard to the order in which they are selected. It is an important tool in probability, statistics, and combinatorial analysis, and is used in a wide variety of fields, including engineering, computer science, and finance. In this essay, we will explore the concept of combination, its properties, and applications.

The combination is defined as the number of ways in which r objects can be selected from a set of n distinct objects, without regard to their order. It is denoted by the symbol C(n,r), which is read as “n choose r.” The formula for calculating the combination is given by:

C(n,r) = n! / (r!(n-r)!)

where “!” denotes the factorial function. The factorial of a positive integer n, denoted by n!, is the product of all positive integers up to and including n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

The combination formula can be derived using the principle of counting, which states that the number of ways in which a task can be performed is equal to the product of the number of ways in which each step of the task can be performed. For example, if we have 5 objects and we want to select 3 of them, the number of ways in which we can do this is equal to the product of the number of ways in which we can choose the first object (5), the number of ways in which we can choose the second object (4), and the number of ways in which we can choose the third object (3). Thus, the total number of ways in which we can select 3 objects from a set of 5 objects is equal to 5 × 4 × 3 = 60. However, this formula counts the same combination multiple times, since the order in which the objects are selected does not matter. To correct for this, we need to divide the result by the number of ways in which the objects can be arranged, which is equal to r! (the factorial of the number of objects selected).

Properties of Combination:

1. The order of the objects does not matter: The combination counts the number of ways in which we can select r objects from a set of n objects, without regard to their order. For example, the combination of 3 objects from the set {1, 2, 3, 4} is the same as the combination of 3 objects from the set {3, 2, 1, 4}.
2. The number of combinations is less than or equal to the number of permutations: The combination of r objects from a set of n objects is always less than or equal to the permutation of r objects from the same set. The permutation counts the number of ways in which we can select r objects from a set of n objects, with regard to their order.
3. The number of combinations is symmetric: The combination of r objects from a set of n objects is equal to the combination of n-r objects from the same set. This property can be proved using the formula for the combination.

Definition of Combination

A combination is a selection of elements from a larger set without regard to the order in which they are selected. This means that the same combination can be obtained in different ways by selecting the elements in a different order. The number of combinations that can be formed from a set of n elements taken r at a time is denoted by the symbol C(n,r) or sometimes denoted as nCr. The formula for calculating C(n,r) is:

C(n,r) = n! / (r! * (n-r)!)

Where n! (n factorial) is the product of all positive integers up to n, and r! (r factorial) is the product of all positive integers up to r. The denominator (r! * (n-r)!) represents the number of ways that the r elements can be arranged among themselves.

Examples of Combination

Example 1: Selecting a team

Suppose you are the manager of a soccer team and you need to select 11 players from a squad of 20 players. How many different teams can you select?

Solution:

The number of ways to select 11 players from a group of 20 is C(20,11), which is calculated as follows:

C(20,11) = 20! / (11! * (20-11)!) = 167,960

Therefore, there are 167,960 different teams that can be selected.

Example 2: Poker hands

In poker, a hand is a combination of five cards. How many different hands can be formed from a standard deck of 52 cards?

Solution:

The number of ways to select 5 cards from a deck of 52 cards is C(52,5), which is calculated as follows:

C(52,5) = 52! / (5! * (52-5)!) = 2,598,960

Therefore, there are 2,598,960 different hands that can be formed from a standard deck of 52 cards.

Example 3: Arranging books on a shelf

Suppose you have 7 books and you want to arrange them on a shelf. How many different ways can you arrange them?

Solution:

The number of ways to arrange 7 books on a shelf is 7!, which is calculated as follows:

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040

However, since the order of the books on the shelf does not matter, we need to divide the result by the number of ways the books can be arranged among themselves. The number of ways that 7 books can be arranged among themselves is 7!, which is calculated as follows:

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040

Therefore, the number of ways to arrange 7 books on a shelf without regard to the order is C(7,7) = 1.

Quiz

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