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Transcript
2.1 Factorial Notation (Textbook Section 4.6) Warm – Up Question How many four-digit numbers can be made using the numbers 1, 2, 3, & 4? (all numbers must only be used once for each 4-digit number) Fundamental Principle of Counting If one operation can be done in “m” ways and another operation can be done in “n” ways, then together, they can be done in mxn ways This principle can be extended to any number of operations i.e. if operation A can be done 3 ways, operation B can be done 4 ways, operation C can be done in 2 ways and operation D can be done 7 ways, then together they can be done in 3 x 4 x 2 x 7 = 168 ways Back to Warm Up Question How many ways can we place the number 1? 1 can be the 1st, 2nd, 3rd or 4th digit, so 4 ways IF we have placed the number 1, how many ways can we place the number 2? One of the digits has already been taken up by the number 1, so there are 3 remaining digit places to put the number 2, so 3 ways Back to Warm Up Question (Continued) IF we have placed numbers 1 and 2, how many ways can we place the number 3? IF we have placed numbers 1, 2, and 3, how many ways can we place the number 4? 2 remaining digit places, so 2 ways One spot remaining, place 1 there, so 1 choice How many ways to place all 4 digits? 4 x 3 x 2 x 1 = 24 ways Factorial Notation Many counting and probability calculations involve the product of a series of consecutive integers (i.e. 4x3x2x1) We can write these products using Factorial Notation The symbol for this notation is: n! or x! How to Use Factorial Notation For all natural numbers (integers > 0) n! represents the product of all natural numbers less than or equal to n n! = n x (n-1) x (n-2) … x 3 x 2 x 1 i.e. 5! = 5 x 4 x 3 x 2 x 1 = 120 Rules for Factorial Notation 0! =1 n!/n! = 1 n!/0! = n!/1 = n!