7.1 Counting Principle, Permutations, Combinations.notebook

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7.1 The Counting Principle, Permutations,

and Combinations

How many possible combinations are there for ordering ice cream?

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Application of the counting principle and permutations

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The fundamental counting principle: Two or more events­ If one event can occur in m ways and another can occur in n ways then the the number of ways that both events can occur is m*n. This can be extended to additional events.

7.1 Counting Principle, Permutations, Combinations.notebook

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2(3)=6

If you have 2 sweaters and 3 pairs of pants/shorts there are possibilities

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The standard Alaskan license plate contains 3 letters followed by 3 numbers.

a) How many different plates are possible if letters and digits can be repeated?

b) How many different plates are possible if letters and numbers cannot be repeated?

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Solution:

a)

b)

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How many phone numbers are possible?

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Permutations­ An ordering of objects. Order is important!

How many permutations are there for the 4 flags shown?

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Use the fundamental counting principle to do this quickly.How many ways are there to choose the first flag?

How many ways are there to choose the second flag? How many ways are there to choose the third flag?

How many ways are there to choose the fourth flag?

7.1 Counting Principle, Permutations, Combinations.notebook

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4*3*2*1= 24

Solution:

This expression can be written as 4!

In general, n! means:

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Permutations of n objects taken r at a time:

The number of permutations of r objects taken from a group of n distinct objects is denoted nPr and is given by the formula:

nPr=

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How many ways can the sophomore class select a president, vice president, and a secretary from a group of 42 people?

First think...does order matter?

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Example: You are burning a demo CD for your band. Your band has 12 songs stored on your computer. However, you only want to put 4 songs on the demo CD. In how many orders can you burn 4 of the 12 songs onto the CD?

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Permutions with Repetition

The number of distinguishable permutations of n objects where one object is repeated S1 times, another is repeated S2 times, and so on is:

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Example:

Find the number of distinguishable permutations of the letters in MISSISSIPPI.

There are 11 letters in the word of which I, and S are repeated 4 times, and P is repeated 2 times. So the number of distinguishable permutations is:

7.1 Counting Principle, Permutations, Combinations.notebook

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Combinations:A combination is a grouping of items in which order does not matter. There are generally fewer ways to select items when order does not matter. For example, there are 6 ways to order 3 items, but they are all the same combinations.

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Tasha deals 5 cards from a deck of 52. How many possible hands are there?

First think...does order matter?

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