MTH 245: Mathematics for Management, Life,and Social Sciences

Boris IskraDepartment of Mathematics. Oregon State University

Section 5.4

Section 5.4:The multiplication principle

Multiplication principle

Suppose that a task is composed of two consecutive choices. If choice 1 canbe performed in m ways and, for each of these, choice 2 can be performed inn ways, then the complete task can be performed in m ·n ways.

Corollary: Extension of multiplication rule

Suppose that a task consists of p choices performed consecutively . If choice1 can be performed in n1 ways and;for each of these, choice 2 can beperformed in n2 ways; for each of these, choice 3 can be performed in n3ways; and so forth. Then the complete task can be performed inn1 ·n2 ·n3 · · ·np ways.

Example 1

A small restaurant offers a special menu from Monday through Friday. Theyhave two main dishes, one with steak and other with mustard chicken. Threedifferent sides: mashed potatoes, French fries, and greens. For dessert: breadpudding and blackberry cobbler.

Example (with tree diagram)

Steak

Potato

Chicken

French Fries

Greens

Potato

French Fries

Greens

Pudding

Cobler

Pudding

Cobler

Pudding

Cobler

Pudding

Cobler

Pudding

Cobler

Pudding

Cobler

Example 2

Standard deck of cards has 4 suits (clubs, spades, hearts and diamonds) of 13denominations each (numbers 2-10 and a Jack, Queen, King, Ace) for a totalof 52 cards.

How many ways are there to choose an Ace and a King from a deck ofcards?

4 ·4 = 16

How many ways are there to choose a Jack, a Queen and a King from adeck of cards?

4 ·4 ·4 = 64

How many ways are there to choose a Diamond and a Spade from adeck of cards?

13 ·13 = 169

Example 3

Licence plate numbers consist of 3 digits followed by 3 letters.How many possible plates are there if you are allowed to repeat digitsand/or letters?

10 ·10 ·10 ·26 ·26 ·26 = 17,576,000

How many possible plates are there if you are not allowed to repeatdigits or letters?

10 ·9 ·8 ·26 ·25 ·24 = 11,232,000

Example 4

Two basketball teams, the “Portland Trail Blazers” and the “DenverNuggets”, have an amicable encounter. There are five players on each teamon the playing court. Once the game is over, they want a picture together,”Portland Trail Blazers” on one side and “Denver Nuggets” on the other.How many different arrangements are possible?

(5 ·4 ·3 ·2 ·1) · (5 ·4 ·3 ·2 ·1) ·2 = 28,800

Example 5

One thousand passengers are traveling from Portland to San Francisco, eachone leaving some day on October and returning to Portland some day onNovember. Claim: At least two passengers are traveling to San Francisco thesame day on October and returning to Portland the same day on November.Why is that true?

31 ·30 = 930

Example 6

In how many ways could 7 people be put in a single line for a photo?

7 ·6 ·5 ·4 ·3 ·2 ·1 = 5,040

Example 7

A painting company has a contract for painting four houses on WashingtonAve at 4:40pm on Friday. The company counts with a total of ten employeesto perform the task. They will choose four employees to do the job (just onefor each house). How many different ways can the company matchemployees with houses?

10 ·9 ·8 ·7 = 5,040

Example 8

How many 4-letter code words are possible using the first 14 letters of thealphabet if

1 no letter can be repeated?2 letters can be repeated?3 adjacent letters cannot be alike?

1 14 ·13 ·12 ·11 = 24,0242 14 ·14 ·14 ·14 = 38,4163 14 ·13 ·13 ·13 = 30,758

Example 9

Suppose that I want to purchase a tablet computer. I can choose either alarge or a small screen; a 64GB, 128GB, or 256GB storage capacity, and ablack or white cover. How many different options do I have?

2 ·3 ·2 = 12

Example 10

Mark cant remember his complete password for his computer account. If heknows only the following information, how many possible passwords will hehave to try?

The password is either 6 or 7 symbols long.It is composed from the letters of the alphabet and the digits 0 through9.It is case-sensitive.The first two symbols must be letters.Mark is positive the last two symbols are 0 and 7, in that order

(52 ·52 ·62 ·62 ·1 ·1)+(52 ·52 ·62 ·62 ·62 ·1 ·1) = 654,833,088

Factorial

We denote by n! (read “n factorial”) the product of all positive integers fromn down to 1:

n! = n · (n−1) · (n−2) · · · · ·2 ·1.

Example 11

Compute 5!

= 5 ·4 ·3 ·2 ·1 = 120