Understanding Permutations and Combinations in Mathematics
Explore the concepts of permutations and combinations, key principles in combinatorial mathematics. A permutation is an arrangement of items in a specific order, while a combination focuses on the selection of items without regard to order. Discover how to calculate the number of ways to arrange or select items using factorials, such as finding the number of ways to line up books or vote for candidates. Examples and detailed explanations illustrate these important mathematical tools, essential for solving problems in probability and statistics.
Understanding Permutations and Combinations in Mathematics
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Presentation Transcript
Def: A permutation is an arrangement of items in a particular order. Ex. How many ways can 10 books be line up on a shelf? Answer: 10*9*8*7*6*5*4*3*2*1 = 3,628,800 Def: n factorial For any positive integer n, n!= n*(n-1)*(n-2) … 3*2*1 For n=0, n!=1.
Number of Permuations The number of permutations of n items of a set arranged r items at a time nPr. nPr = n! / (n-r)! Ex. 10P4 = 10! / (10-4)! = 10! / 6! = 5040
Example: Seven yachts enter a race. First, second, and third will be given to the three fastest yachts. How many ways and the yachts finish? Answer: 7P3 = 7! /(7-3)! = 7! / 4! = 210
Def: A selection in which order does notmatter is a combination. Number of Combinations The number of combinations of n items of a set chosen r items at a time is nCr. nCr = n! / r!*(n-r)! Example: 5C3= 5! / 3!(5-3)! = 120/6*2 = 10
Example: Ten candidates are running for three seats in the student government. You may vote for as many as three candidates. In how many ways can you vote for three or fewer candidates? Answer: This is a combination so we can vote for 3 people, or 2 people, or 1 person, or none one.
3 people: 10C3 120 ways. 2 people: 10C2 45 ways 1 person: 10C1 10 ways No one: 10C0 1 way
