In high school, permutations and combinations are emphasized in Integrated Math II (or Algebra II) and the Math Analysis (precalculus) courses. The Fundamental Counting Principle is introduced in elementary and middle school and forms the foundation for enumerating quantities given varying choices. Counting allows us to enumerate all possible ways an event can occur, and from these counts we can make inferences that will inform our decision making. People use basic principles of combinatorics in many everyday situations. Events are the set of all possible outcomes resulting from a particular situation, experiment, or process. When was the last time you were in a situation where you thought to yourself, “There are too many options, I can’t decide what I want!” or “What are the chances?” Nearly all situations can be quantified using methods or concepts in combinatorics, the branch of mathematics that includes selections and arrangements of objects with prescribed conditions and configurations. Other decisions are more complicated and may involve breaking a situation down into smaller choices. Some of these decisions are only two options, being ‘yes’ or ‘no’. It has been estimated that the average adult makes 35,000 decisions per day (7). An application of this relationship is in determining the number of possible energy states in an energy distribution for distinguishable particles.Enumerating Daily Life with Counting Principles, Permutations, and Combinations by Lawrence E. Here the terms in the denominator are the populations of the k subsets. This quantity nC r is called the "combination" and represents the probability of picking r distinguishable outcomes out of n without regard to the order of picking each outcome.įor a group of n objects or events which are broken up into k subsets, the above relationship is generalizable to give the number of distinguishable permutations of the n objects. The number of distinguishable collections of r objects chosen from n is obtained by dividing the permutation relationship by r!. The factor of overcounting in this case is 6 = 3!, the number of permutations of 3 objects. So the permutation relationship overcounts the number of ways to choose this combination if you don't want to make a distinction between them based on the order in which they were chosen. For the purposes of card playing, the following ways of drawing 3 cards are equivalent: The number of permutations of r objects out of n is sometimes what you need, but it has the drawback of overcounting if you are interested in the number of ways to get distinguishable collections of objects or events. Which can be expressed in the standard form For the first pick, you have n choices, then n-1 and so on down to n-r+1 for the last pick. Now if you are going to pick a subset r out of the total number of objects n, like drawing 5 cards from a deck of 52, then a counting process can tell you the number of different ways you can do that. In general we say that there are n! permutations of n objects. So we say that there are 5 factorial = 5! = 5x4x3x2x1 = 120 ways to arrange five objects. Note that your choice of 5 objects can take any order whatsoever, because your choice each time can be any of the remaining objects. As illustrated before for 5 objects, the number of ways to pick from 5 objects is 5!. On your second pick, you have n-1 choices, n-2 for your third choice and so forth. If you are making choices from n objects, then on your first pick you have n choices. The permutation relationship gives you the number of ways you can choose r objects or events out of a collection of n objects or events.Īs in all of basic probability, the relationships come from counting the number of ways specific things can happen, and comparing that number to the total number of possibilities. The number of tennis matches is then the combination. If you don't want to take into account the different permutations of the elements, then you must divide the above expression by the number of permutations of r which is r!. So in only 15 matches you could produce all distinguishable pairings. If you have a collection of n distinguishable objects, then the number of ways you can pick a number r of them (r < n) is given by the permutation relationship:įor example if you have six persons for tennis, then the number of pairings for singles tennis isīut this really double counts, because it treats the a:b match as distinct from the b:a match for players a and b. Permutations Permutations and Combinations
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