3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia.

How many different

routes can you take

for the trip to

Philadelphia by way

of Trenton?

________ • _________

Trenton Philadelphia

___4____ • ___3_____

12

4. You have 10 pairs of pants, 6 shirts, and 3 jackets.

How many outfits

can you have

consisting of a

shirt, a pair of

pants, and a

jacket?

______•______•______

Shirts Pants Jackets

___6__•__10__•__3___

180

5. Fifteen people line up for concert tickets. a) How many

different

arrangements are

possible?

15•14•13•12•11•10•9•8•

7•6•5•4•3•2•1 =

1,307,674,368,000

b) Suppose that a

certain person must

be first and another

person must be last.

How many

arrangements are now

possible?

1•13•12•11•10•9•8•

7•6•5•4•3•2•1•1 =

6,227,020,800

6) Using the letters A, B, C, D, E, Fa) How many “words”can be made using all 6letters?6 • 5 • 4 • 3 • 2 • 1 = 720b) How many of thesewords begin with E ?1 • 5 • 4 • 3 • 2 • 1 = 120c) How many of thesewords do NOT beginwith E? 720 –120 = 600d) How many 4-letterwords can be made ifno repetition is allowed?6•5•4•3 = 360

e) How many 3-letterwords can be made ifrepetition is allowed?6 • 6 • 6 = 216f) How many 2 OR 3letter words can bemade if repetition isnot allowed? 6•5+6•5•4 = 30 + 120 = 150g) If no repetition isallowed, how manywords containing atleast 5 letters can bemade? (both letter 6a)720 + 720 = 1440

16.3 Distinguishable Permutations

OBJ: To find the quotient of numbers given in factorial notation

To find the number of distinguishable permutations when some of the objects in an arrangement are alike

EX: Find the value of 8! _ 4! x 3!

One Method

8 • 7 • 6 • 5 • 4 • 3 • 2 • 1

4 • 3 • 2 • 1 • 3 • 2 • 1

Short Method

8 • 7 • 6 • 5 • 4!4! • 3 • 2 • 11680 6280

EX: Find the value of 6! _ 4! x 2!

Short Method

6 • 5 • 4!4! • 2 • 130 2

15

EX: Find the value of 12! _ 3! x 9!

Short Method

12 • 11 • 10 • 9!3 • 2 • 1 • 9!1320 6

220

NOTE: The letters in the word Pop are distinguishable since one of the two p’s is a capital letter. There are 3!, or 6, distinguishable permutations of P, o, p.

Pop Ppo oPp opP poP pPo

In the word pop, the two p’s are alike and can be permuted in 2! ways. The number of distinguishable permutations of p, o, p is 3! , or 3.

2!

pop ppo opp

The number of distinguishable permutations of the 5 letters in daddy is 5! 3! since the three d’sare alike and can be permuted in 3! ways.

DEF: Number of Distinguishable PermutationsGiven n objects in which a of them are

alike, the number of distinguishable

permutations of the n objects is n!

a!

EX: How many distinguishable permutations can be formed from the six letters in pepper? 6!__

3! • 2!

6 • 5 • 4 • 3!3! • 2 • 1

60

EX: How many distinguishable six- digit numbers can be formed from the digits of 747457? 6!__

3! • 2!

6 • 5 • 4 • 3!3! • 2 • 1

60

EX: How many distinguishable signals can be formed by displaying eleven flags if 3 of the flags are red, 5 are green, 2 are yellow, and 1 is white? 11!______

3! • 5! • 2! • 1!11 • 10 • 9 • 8 • 7 •6 •5!3 • 2 • 1 •5! • 2 •1 •1332640 1227720

16.4 Circular Permutations

OBJ: To find the number of possible permutations of objects in a circle

NOTE: Three objects may be arranged in a line in 3!, or 6, ways. Any one of the objects may be placed in the first positionABC ACB BAC BCA CAB CBA

In a circular permutation of objects, there is no first position. Only the positions of the objects relative to one another are considered.EX: In the figures below, Al, Betty and

Carl are seated in a circular position with

each person facing the center of the

circle.

In each of the first three figures, Al has Betty to his left and Carl to his right. This is one circular permutation of Al, Betty, and Carl.

A

C B

C

B A

B

A C

The remaining three figures each show Al with Betty to his right and Carl to his left. Again, these count as only one circular permutation of the three

A

B C

B

C A

C

A B

DEF: Number of Circular Permutations

The number of circular permutations of n distinct objects is (n-1)!

EX: A married couple invites 3 other couples to an anniversary dinner. In how many different ways can all of the 8 people be seated around a circular table?(8 – 1)!

7!

5040

7. How many distinguishable permutations can be made using

all the letters of:a) GREAT

5 • 4 • 3 • 2 • 1

5! = 120

b) FOOD

4! = 4 • 3 • 2!

2! 2!

12

c) TENNESSEE

9!

4! 2! 2!1!

9 • 8 • 7 • 6 • 5 • 4!

4! 2 • 2

15,120

4

= 3,780