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Understanding Permutations and Combinations: Counting Principles in Mathematics

This tutorial introduces the fundamental counting principles of permutations and combinations. Learn how to calculate the number of ways events can occur using the Fundamental Counting Principle. Explore factorial notation and its significance in counting theory. Discover the formulas for permutations (nPr) and combinations (nCr) through practical examples, such as selecting officers from a group and calculating lottery combinations. Enhance your understanding of these concepts to solve various probability problems effectively.

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Understanding Permutations and Combinations: Counting Principles in Mathematics

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  1. PERMUTATIONS AND COMBINATIONS Brought to you bytutorial services – the math center

  2. Fundamental Counting Principle If there are n(A) ways in which an event A can occur, and if there are n(B) ways in which a second event B can occur after the first event has occurred, then the two events can occur in n(A) · n(B) ways. Special products such as 4! (4 · 3 · 2 · 1) (or any other constant) frequently occur in counting theory. This symbol is a special notation, known as factorial. Factorial is explained as: For any positive integer n, we define n-factorial, written as n! = n(n - 1)(n - 2)(n - 3)……….. We define 0! =1

  3. Permutation and Combination Formulas • Permutation - The number of possible distinct arrangements of r objects chosen from a set of n objects is called the number of permutations of n objects taken r at a time and it equals: nPr = __n!__ (n – r)!

  4. Permutation and Combination Formulas • ExampleIn how many ways can a president, vice president, secretary, and treasurer be selected from an organization with 20 members? • Solution (the number of arrangements in which 4 people can be selected from a group of 20)n = 20 r = 4 nPr = 20!__ = 20 · 19 · 18 · 17 · 16! = 116,280 (20 - 4)! 16!

  5. Permutation and Combination Formulas • Combination - The number of combinations of n objects taken r at a time is: nCr = ___n!___ r!(n – r)!

  6. Permutation and Combination Formulas • ExampleIn the Texas lottery you choose 6 numbers from 1 though 54. If there is no replacement or repetition of numbers, how many different combinations can you make? • Solutionn = 54 r = 6 • nCr = 54!__ = 54 · 53 · 52 · 51 · 50 · 49 = 25,827,165 • 6! (54-6)! 720

  7. Permutations and Combinations Links • Probability Handout • Probability Workshop

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