The Fundamental Counting Principle The fundamental counting principle

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The Fundamental Counting Principle

The Fundamental Counting Principle

The fundamental counting principle is the primary rule for calculating the number of possible

The fundamental counting principle is the primary rule for calculating the number of possible outcomes.

FOR EXAMPLE: Pretend that you are putting together an outfit. You have 3 pairs

FOR EXAMPLE: Pretend that you are putting together an outfit. You have 3 pairs of pants (blue, black, and tan), 2 shirts (white and red), and 2 hats (green and grey). How many different outfits can you make?

So there Start with thethis Pants… Then youare can choose The possible manyshirts possible

So there Start with thethis Pants… Then youare can choose The possible manyshirts possible outfits. Three Choices are… Two hats combinations 1 Blue – Red - Green 2 Blue – Red - Gray 3 Blue – White - Green 4 Blue – White - Gray 5 Black – Red - Green 6 Black – Red - Gray 7 Black – White - Green 8 Black – White - Gray Tan – Red - Green 9 Tan – Red - Gray Tan – White - Green Tan – White - Gray 10 11 12

EXAMPLE Make a tree diagram to represent the following situation: You are making a

EXAMPLE Make a tree diagram to represent the following situation: You are making a sandwich. You have 2 breads (white and wheat), 3 meats (ham, turkey, and roast beef) and 3 cheeses (cheddar, swiss, and American). How many sandwiches can you make with one bread, one meat, and one cheese?

THE COUNTING PRINCIPLE ·A more efficient way of counting is necessary to handle large

THE COUNTING PRINCIPLE ·A more efficient way of counting is necessary to handle large masses of statistical data (e. g. the level of inventory at the end of a given month, or the number of production runs on a given machine in a 24 hour period, etc. ), and for an understanding of probability. ·It is not practical to make a tree diagram for every combination of events.

THE MULTIPLICATION COUNTING PRINCIPLE ·If one event can occur in m ways and another

THE MULTIPLICATION COUNTING PRINCIPLE ·If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur together is m x n. ·This principle applies given any number of events and can be used to efficiently determine the number of outcomes. ·For example, given 2 breads, 3 cheeses and 2 meat options, the total number of sandwiches we can make is 2 x 3 x 2 = 12.

Some Notes on THE COUNTING PRINCIPLE When applying the counting principle(s), be sure to

Some Notes on THE COUNTING PRINCIPLE When applying the counting principle(s), be sure to take into account all possibilities. This includes the positions of the items and how many items fit that position. For example, the first three digits of a phone number (after the area code) are called the prefix and are governed by the following rule. The first digit cannot be zero or one because those numbers dial the operator or indicate long distance. How many 3 -digit prefixes exist for US phone numbers? 8 10 10 No. of Choices? Number of Possible Prefixes: } 8 x 10 = 800

Example: Confirmation Codes Every purchase of an i-pad made on Apple’s website is given

Example: Confirmation Codes Every purchase of an i-pad made on Apple’s website is given a randomly generated confirmation code. The code consists of 3 symbols (letters and digits). How many different codes can be generated if at least one letter is used in each code? Think of the possibilities… Using only 1 letter… Where could the letter be in the code? Using 2 letters… Where could the letters be in the code? Using 3 letters… There is only one way for this to happen…

Codes with ONE Letter Codes with TWO Letters Codes with THREE Letters 26 26

Codes with ONE Letter Codes with TWO Letters Codes with THREE Letters 26 26 26 10 The letter can be In 1 st, 2 nd, or 3 rd… 26 x 10 x 3 = 7, 800 10 The letters can be in 1 st & 2 nd, or 1 st & 3 rd or 2 nd & 3 rd 26 x 10 x 3 = 20, 280 26 The letters have to Take up all three positions 26 x 1 = 17, 576 TOTAL NUMBER OF CODES: 7, 800 + 20, 280 + 17, 576 = 45, 656