combinations介绍英文版PPT
Introduction to CombinationsCombinations are a fundamental concept in mathema...
Introduction to CombinationsCombinations are a fundamental concept in mathematics, particularly in the fields of combinatorics and probability theory. They refer to the number of ways in which a set of distinct items can be selected, regardless of the order in which they are selected. In contrast to permutations, which consider the order of selection, combinations focus on the number of unique subsets that can be formed from a given set.In this article, we will explore the concept of combinations in detail, including their definition, properties, and applications. We will also provide examples and formulas to illustrate the calculations involved in finding combinations.Definition of CombinationsA combination is a selection of objects from a set, where the order of selection does not matter. More formally, a combination of (n) distinct objects taken (r) at a time is an unordered collection of (r) objects from the set of (n) objects.The number of combinations of (n) distinct objects taken (r) at a time is denoted by (C(n, r)) or (\binom{n}{r}). It is given by the formula:[ C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} ]where (n!) denotes the factorial of (n), which is the product of all positive integers less than or equal to (n).This formula can be interpreted as follows: there are (n) ways to choose the first object, (n-1) ways to choose the second object (since one object has already been selected), and so on, until there are (n-r+1) ways to choose the (r)th object. The total number of combinations is obtained by multiplying these choices together and dividing by the number of permutations of the (r) selected objects, which is (r!).Properties of CombinationsCombinations exhibit several important properties that are useful in understanding their behavior and applications. Some key properties of combinations are listed below:The sum of the number of combinations of (n) distinct objects taken (r) at a time and the number of combinations taken (n-r) at a time is equal to the number of combinations taken (n) at a time[ C(n, r) + C(n, n-r) = C(n, n) ]This property arises from the fact that selecting (r) objects from a set of (n) objects is equivalent to selecting the remaining (n-r) objects.The number of combinations of (n) distinct objects taken (r) at a time is equal to the number of combinations taken (r) at a time from (n) distinct objects[ C(n, r) = C(n, n-r) ]This property follows from the complementary property and the fact that selecting (r) objects from a set leaves (n-r) objects unselected.The number of combinations of (n) distinct objects taken (r) at a time is equal to the sum of the number of combinations taken (r-1) at a time and the number of combinations taken (r) at a time from (n-1) distinct objects[ C(n, r) = C(n-1, r-1) + C(n-1, r) ]This identity provides a recursive relation for calculating combinations and is the basis for the Pascal's triangle, a triangular array of numbers that illustrates the relationships between combinations.The number of combinations of (n) distinct objects taken 0 or 1 at a time is given by[ C(n, 0) = 1 ][ C(n, 1) = n ]These properties provide a foundation for understanding and applying combinations in various mathematical and practical contexts.Applications of CombinationsCombinations have a wide range of applications in various fields, including combinatorics, probability theory, statistics, computer science, and more. Some common applications of combinations are described below:Probability and StatisticsCombinations play a crucial role in probability and statistics, particularly in calculating the number of possible outcomes in experiments or events. For example, in a lottery game with (n) distinct numbers and a player selecting (r) numbers, the probability of winning is determined by the ratio of the number of combinations of (r) numbers from (n) to the total number of combinationsCounting ProblemsCombinations are often used in counting problems, where the goal is to determine the number of ways in which
