counting principles and probability digital lesson

Counting Principles and Probability

Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

The Fundamental Counting Principle states that if one event can occur m ways and a second event can occur n ways, the number of ways the two events can occur in sequence is m • n.

1st Coin Tossed

Start

Heads Tails

Heads Tails

2nd Coin Tossed

There are 2 2 different outcomes: {HH, HT, TH, TT}.

Heads Tails

2 ways to flip the coin.

2 ways to flip the coin.


Example: A meal consists of a main dish, a side dish, and a dessert. How many different meals can be selected if there are 4 main dishes, 2 side dishes and 5 desserts available?

# of main dishes

# of side dishes

# of desserts

4 52 =

There are 40 meals available.

40


A permutation is an ordered arrangement of n different elements.

How many permutations are possible using the three colors red, white, and blue?

There are 3 choices for the first color, 2 choices for the second color and only 1 choice for the third color.

3! = 3 • 2 • 1 = 6 permutations

“factorial”


__ x __ x __

A permutation of n elements taken r at a time is a subset of the collection of elements where order is important.

Five projects are entered in a science contest. In how many ways can the projects come in first, second, and third?

35 projects

5 4

5 • 4 • 3 = 60 ways

4 projects

3 projects1st 2nd 3rd


The formula for the number of permutations of n elements taken r at a time is

5 3n rP P 5! 5 4 3 2 12! 2 1

60

n rP# in the

collection # taken from the

collection

! .( )!

nn r


The T’s are not distinguishable.

STATS

If some of the items are identical, distinguishable permutations must be used.

In how many distinguishable ways can the letters STATS be written?

STATS

The S’s are not distinguishable.

Example continues.


The number of distinguishable permutations of the n objects is

1 2 3

!! ! ! !k

nn n n n

The letters STATS can be written in

5!2! 2! 1!

S’s T’s A’s

120 30 ways.4

where n = n1 + n2 + n3 + . . . + nk.


A combination of n elements taken r at a time is a subset of the collection of elements where order is not important.

Using the letters A, B, C, and D, find all the possible combinations using two of the letters.

{AB}{AC}{AD}{BC}{BD}{CD}

This is the same as {BA}.

There are six different combinations using 2 of the 4 letters.


! .( )! !

nn r r

4 24!

2!2! n rC C

The formula for the number of combinations of n elements taken r at a time is

n rC# in the

collection # taken from the

collection

4 3 2 12 1 2 1 6


Example: How many different ways are there to choose 6 out of 10 books if the order does not matter?

10 6 n rC C

There are 210 ways to choose the 6 books.

10 9 8 7 6!4 3 2 1 6!

3

21010!4!6!


Graphing Utility: Permutation

5 3P

10 6C

Graphing Utility: Combination

counting principles and probability digital lesson

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