Introduction to Combinatorics Section 2.3. The Fundamental Principle of Counting The total number of possible outcomes of a series of decisions (i.e.

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Presentation transcript:

Introduction to Combinatorics Section 2.3

The Fundamental Principle of Counting The total number of possible outcomes of a series of decisions (i.e. making selections from various categories) is found by multiplying the number of choices for each decision (or category). Example1: The Pick – 3 Lottery. –How many possible Pick – 3 numbers are there? (You can repeat digits). –Each digit in the Pick – 3 ranges from 0 to 9, thus there are 10 digits to choose from. –Therefore, there 10 x 10 x 10 = 1000 Pick – 3 numbers.

Example 2: Arizona License plates How many possible Arizona license plates are there given that the first 3 places on the plate are letters (Caps only) and the last 3 places are digits (0 thru 9)? Answer: Since there are 6 places on a AZ plate we have 6 categories. 3 categories are letters 3 are digits. Since there are 26 letters and 10 digits the answer is: 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000.

Example 3: Number of rows in a truth table. Each simple statement has 2 choices true or false. Let’s say I have one simple statement, then that statement is true or false, hence a truth table with 2 rows. If I have two simple statements then each statement can be true or false so the number of rows would be 2 x 2 = 4 If I have three simple statements then the number of rows is 2 x 2 x 2 = 8

Factorials Factorials are a short-hand way of writing out a string of multiplications. The symbol for factorial is the exclamation point ! Five factorial is 5! = 5 x 4 x 3 x 2 x 1 = 120 If n is a positive integer, then n factorial is defined as n! = n x (n – 1) x (n – 2) … x 2 x 1. As a special case 0! = 1.

Examples of factorials 0! = 1 1! = 1 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6 4! = 4 x 3 x 2 x 1 = 24 5! = 5 x 4 x 3 x 2 x 1 = 120 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800 Wow! They start out slow but get big very fast.

Factorial Arithmetic Compute the following: 1.(7!)(3!) = (5040)(6) = 30,240 Note this is NOT 21! = 51,090,942,171,709,440,000. I believe you would say this as 51 quintillion, 90 quadrillion, 924 trillion, 171 billion, 709 million, 440 thousand. 2. Notice that you get some cancellation. You are left with