3. suppose you take 4 different routes to trenton, the 3 different routes to philadelphia
DESCRIPTION
How many different routes can you take for the trip to Philadelphia by way of Trenton?. ________ • _________ Trenton Philadelphia ___ 4 ____ • ___3_____ 12. 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia. How many outfits can you have - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/1.jpg)
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
![Page 2: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/2.jpg)
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
![Page 3: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/3.jpg)
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
![Page 4: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/4.jpg)
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
![Page 5: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/5.jpg)
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
![Page 6: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/6.jpg)
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
![Page 7: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/7.jpg)
EX: Find the value of 6! _ 4! x 2!
Short Method
6 • 5 • 4!4! • 2 • 130 2
15
![Page 8: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/8.jpg)
EX: Find the value of 12! _ 3! x 9!
Short Method
12 • 11 • 10 • 9!3 • 2 • 1 • 9!1320 6
220
![Page 9: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/9.jpg)
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
![Page 10: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/10.jpg)
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
![Page 11: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/11.jpg)
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.
![Page 12: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/12.jpg)
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!
![Page 13: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/13.jpg)
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
![Page 14: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/14.jpg)
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
![Page 15: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/15.jpg)
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
![Page 16: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/16.jpg)
16.4 Circular Permutations
OBJ: To find the number of possible permutations of objects in a circle
![Page 17: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/17.jpg)
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
![Page 18: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/18.jpg)
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.
![Page 19: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/19.jpg)
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
![Page 20: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/20.jpg)
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
![Page 21: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/21.jpg)
DEF: Number of Circular Permutations
The number of circular permutations of n distinct objects is (n-1)!
![Page 22: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/22.jpg)
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
![Page 23: 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia](https://reader036.vdocuments.us/reader036/viewer/2022062314/5681389f550346895da05c48/html5/thumbnails/23.jpg)
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