Section 6.3 The Multiplication Principle

The Multiplication Principle

Suppose there are m ways to do a task T1 and there are n ways to do a task T2. Then there are m · nways of doing the task T1 followed by the task T2.

1. Four commuter trains and two express buses depart from city A to City B in the morning, and

five commuter trains and five express buses operate on the return trip in the evening (from City

B to City A). In how many ways can a commuter from City A to City B complete a daily round

trip via bus and/or train?

Generalized Multiplication Principle

Suppose a task T1 can be done in N1 ways, a task T2 can be done in N2 ways,...,and, finally, a

task Tm can be done in Nm ways. Then the number of ways of doing the tasks T1, T2, ..., Tm in

succession is given by the product

N1N2 · · ·Nm.

2. A new state employee is o↵ered a choice of eight basic health plans, five dental plans, and two

vision care plans. How many di↵erent health-care plans are there to choose from if one plan is

selected from each category?

3. In recent years, a state has issued license plates using a combination of two letters of the alphabet

followed by three digits, followed by another two letters of the alphabet. How many di↵erent

license plates can be issued using this configuration?

÷ ⇒ 3×2=6

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-

-T to B

"

# of ways to do Ti followed# of ways to do To = 4+2=6

by Tz = 6 x 10

* III:3.

'

III. rests =60u

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.

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€÷ne of BHP"

→ 8 ways

' ::::::÷:3 's:t÷÷

-26Etototo IEez

=264.103

= 456,976,0€

4. Complete the following.

(a) How many seven-digit telephone numbers are possible if the first digit must be nonzero?

(b) How many international direct-dialing numbers for calls within the United States and Canada

are possible if each number consists of a 1 plus a three-digit area code (the first digit of which

must be nonzero) and a number of the type described in part (a)?

5. How many three-digit numbers can be formed from the numerals in the set {1, 2, 3, 4} if the

following is true?

(a) Repetition of digits is allowed.

(b) Repetition of digits is not allowed.

6. A state makes license plates with three letters followed by four digits.

(a) How many license plates are possible?

(b) If no repetition of the letters is permitted, how many di↵erent license plates are possible?

(c) If no repetition of letters or digits is permitted, how many di↵erent license plates are possible?

2 Fall 2017, Maya Johnson

- -

a-eeEeee

-7g:6%a- -

--

1 9- Etf-9toto-0=92×108

28,100,000,000J-

-

4- =

43=640-

-43--2 =4×3×2=240-

-262-2261EEE-10 =263×104 --175,760,0€

-

-262-52-4/10toto-10=26.25 .24.104

#

= 156,000,0002L

-26-24/10

-98--7=26.25

. 24.10.98 .

7=78,624,00071

(d) How many license plates have no repetition of letters or digits and begin with a vowel?

n-Factorial (n!) For any natural number n,

n! = n(n� 1)(n� 2) · · · 3 · 2 · 1

0! = 1

7. Find 3!, 4! and 7!

8. Five men and ten women are to line up for a picture with the five men in the middle. How many

ways can this be done? (Assume there are five women on each side of the group of men.)

9. An exam consits of three true/false questions followed by four multiple choice questions each with

3 answers.

(a) How many ways can a student answer the exam if they answer all of the questions?

(b) How many ways can a student answer the exam if they can leave true/false questions blank?

(c) How many ways can a student answer the exam if they can leave any of the questions blanks?

3 Fall 2017, Maya Johnson

5 vowels

-525 24 to 9 8 7=5 . 25.24010%8-7- -

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us.ro#y

3 ! -

- 3. 2.1=60 71=7.6 . 5. 4.3 .

2.I

4 ! - 4 . 3. 2.I = ④ =5,

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Women Men Women

to. -9*7-6/-5.4=3.tt/E.4eE#w=onne&!.m%a1=435i4-6,oooJ

-

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-21¥ .-3=31--6480-

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