warm up: 2/21/2012 use completing the square. fundamental counting principles if one selection can...

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Warm Up: 2/21/2012 0 32 8 2 : . 1 2 x x Solve 4 2 x x f , : . 2 1 f Find Use completing the square

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Fundamental Counting Principles  If one selection can be made in m ways, and for each of these a second selection can be made in n ways, then the number of ways the two selections can be made is  If the possibilities being counted can be grouped into mutually exclusive cases, then the total number of possibilities is the sum of the number of possibilities in each case

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Page 1: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

Warm Up: 2/21/2012

03282:.1 2 xxSolve 42 xxf,:.2 1fFindUse completing the square

Page 2: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

15.5 Fundamental Counting Principles 2/21/2012

Page 3: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

Fundamental Counting PrinciplesIf one selection can be made in m ways, and for each of these a second selection can be made in n ways, then the number of ways the two selections can be made is

If the possibilities being counted can be grouped into mutually exclusive cases, then the total number of possibilities is the sum of the number of possibilities in each case

Page 4: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

EXAMPLE #1Step 1: How many choices for model

Step 2: How many choices for color

Step 3: Multiply the number of choices

A local moped dealer sells 6 different models of mopeds. Each model is available in 3 colors. How many combinations of model and color are there?

Page 5: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

Example #2How many odd 2-digit whole numbers less than 70 are there?

Step1: how many choices for the tens’ digit

Step 2: how many choices for the units’ digit

Step 3: multiply the number of choices

Page 6: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

Example #3Step 1: How many choices for blouses

Step 2: How many choices for scarves

Step 3: Multiply the number of choices

Elena can wear one of 2 blouses and one of 5 scarves. How many blouse-scarf combinations are available to her?

Page 7: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

Example #4How many positive integers less than 100 can be written using the digits 6, 7, 8, and 9?

Step 1: number of outcomes for the 1 digit integers

Step 2: number of outcomes for the 2 digit integers

Step 3: Add the outcomes

Page 8: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

Example #5Step 1: Number of

outcomes for 1-letter case

Step 2: Number of outcomes for 2-letter case

Step 3: Number of outcomes for 3-letter case

Step 4: Add the outcomes

How many license plates of 3 symbols (letters and digits) can be made using at least one letter in each?

Page 9: Warm Up: 2/21/2012 Use completing the square. Fundamental Counting Principles If one selection can be made in m ways, and for each of these a second

Example #6How many positive odd integers less than 10,000 can be written using the digits 3, 4, 6, 8, and 0

Step 1: How many 1 digit numbers

Step 2: How many 2-digit numbers

Step 3: How many 3-digit numbers

Step 4: How many 4-digit numbers

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