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Chapter 10

Counting Techniques

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Sequential Counting Principle

Section 10.1

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Tree Diagrams

Tree Diagrams are used to display all the possible outcomes of a situation.

T

H

TH

H

T

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Sequential Counting Principle

SCP

If an event can occur in m ways and a second event can occur in n ways, then the sequence of two events can occur in m x n ways.

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SCP Fundamental Counting Principle

If an event can occur in m ways, and then a second event can occur in n ways, and then a third event can occur in r ways, and so on, then the sequence of events can occur in m n r … ways.

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Exercise:

A bag contains 6 brown marbles and 3 white marbles. You are asked to draw 3 marbles from the bag without replacement. In how many ways can you draw 2 brown and 1 white in that order?

6 x 5 x 3 = 901st B 2nd B 1st W

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Exercise:

In how many ways can 1 woman and 1 man, in that order, be selected from 5 women and 6 men?

5 x 6 = 30

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Exercise:

How many different sets of 3 initials can be made from the letters of the English alphabet?

Hint: AAA is different from BBB.

26 x 26 x 26 = 17,576

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Exercise:

The Good Taste Restaurant has 7 entrees, 6 vegetables, and 9 desserts on its menu. If you want to order 1 entree, 1 vegetable, and 1 dessert, how many choices do you have?

7 x 6 x 9 = 378

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In Connecticut, auto license plates carry 3 digits followed

by 3 letters.

a. How many arrangements are possible for the 3

letters?

b. How many arrangements are possible for the 3

numbers?

c. How many different license plates can be made

using 3 numbers followed by 3 letters?

Exercise:

a. 26 x 26 x 26 = 17,576

b. 10 x 10 x 10 = 1000

c. 17,576 x 1000 = 17,576,000

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Social Security numbers consist of nine digits.

a. If the first digit cannot be a 0, how many Social Security numbers are possible?

b. How many Social Security numbers are possible if there are no restrictions?

Exercise:

9 x 108 = 900,000,000

109 = 1,000,000,000

9 10 10 10 10 10 10 10 10

10 10 10 10 10 10 10 10 10

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DICE

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You are expected to be familiar with a standard deck of playing cards.

Standard deck 52 cards

Black 26 Red 26 13 Spades ♥ 13 Hearts

♣ 13 Clubs ♦ 13 Diamonds

12 Face Cards (Jack, Queen, & King)

Black 6 Red 6 3 Spades ♥ 3 Hearts

♣ 3 Clubs ♦ 3 Diamonds

40 Non - Face Cards (A -10)

Black 20 Red 20 10 Spades ♥ 10 Hearts

♣ 10 Clubs ♦ 10 Diamonds

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Note: the Ace will be considered to be a one. There are four suits: Clubs, Diamonds, Spades, and Hearts in a deck of cards.

END