Chapter 12: Permutations and Combinations

How many wys can a 15 question True-False Quiz be answered if: All but one question is answered?

(2^ 14) times 15* = 245760 * you multiply by 15 b/c ANY ONE question of the 15 can be left blank

How many 4 digit numbers can be made using 0,5,9,8,7 if digits can not be repeated?

*4 4 3 2 * first term is 4 b/c there can't be a 0 there otherwise won't be 4 digit

101! / 99!

101 * 100= 10100

Ward Robe has 15 pairs of slacks and 23 shirts. How many different ways could he select a slacks-and-shirt combo?

15 times 23 = 345

20 girls are on the bb team. 17 are over 16 and 12 are over 170 cm tall, and 9 are both over 16 yrs and over 170 cm tall. How many of the girls are over 16 or over 170 cm tall?

17 + 12 - 9 = 20

Express as ratio of factorials: (15*14) / (20*19)

18! 15! / 20! 13!

How many positive and negative factors does 180 have?

180= 2^2 times 5 times 3^2 3 times 2 ties 3===> 18 *times 2 = 36 * times 2 cuz both positive and negative

A flight flying out of Chicago Airport has flights in London, Paris, Hawaii, Mexico City, Rome, and Tokyo. There are six pilots (P) available. P1 requests to be scheduled to Mexico city and P2 requests Tokyo. How many ways can the remaining pilots be scheduled?

2 out of 6 pilots have a request KEY WORD: Remaining Pilot There are 4 remaining pilots to be scheduled for the other 4 flights, so: 4! = 4 times 3 times 2 times 1 ==> 24

Express as ratio of factorials: 20*19*18

20! / 17!

How many positive factors does 2100 have?

2100= 5^2 times 2^2 times 3 times 7 3 times 3 times 2 times 2===> 36

Express as ratio of factorials: 29*28*27*9*8*7*6

29! / 26! 5!

How many ways can a 15 question True-False Quiz be answered if: Every question is answered?

2^ 15 = 32768

How many positive integral factors does 60,000 have?

2^5 + 5^4 + 3= 60,000 6* 5*2=60

How any subsets of ABCD are there?

2^n = Ans 2^ 4 = 16

The senior class has 367 girls, and 425 students with brown hair. 296 of the girls have brown hair. In how many different ways could you select a girl or a brownhaired student from the senior class?

367+ 425-296= 496

Researchers discover 37 techniques used in working math problems. 29 techniques used working physics problems. 21 used in both fields. If Wanda studies both physics and math, how many different techniques must she learn?

37 + 29 - 21= 45

How many ways can 4 people stand in a line?

4! = 24

The library has 463 books dealing with science and 592 fiction books. Of these, 37 are science fiction books. HOw many books do they have that are science or fiction?

463 + 592 - 37 = 1018

Given 5 different physics book, 4 different math books, and 7 different english books: How many ways are there to select one book?

5 + 4 + 7 = 16

How many 4 digit numbers can be made using digits 4,5,1,2,3 no more than once?

5 4 3 2 ==> 120

How many even 3 digit numbers can be made with 2,4,6,5,9 with repeat of digits allowed?

5 5 3 ==>75

How many 4 digit numbers can be made using 1,2,3,4,5 if digits can be repeated?

5 5 5 5 ==> 625

How many ways can you make a sundae with 5 flavors and 4 different toppings?

5 time 4 = 20

A 6-letter permutation is selected at random from the letters NIMBLE: How many of these permutations begin with M?

5! = 120

A jewelry store has 544 necklaces and 215 pieces containing shells. 129 of the pieces are shell necklaces. How many of the pieces are necklaces or contain shells?

544 + 215 -129 = 630

A flight flying out of Chicago Airport has flights in London, Paris, Hawaii, Mexico City, Rome, and Tokyo. There are six pilots (P) available. How many ways can the pilots be scheduled for the flights if each pilot is scheduled for only one city?

6! 6 time 5 times 4 times 3 times 2 times 1 ==> 720

How many ways can you arrange: 6 books on a shelf?

6! = 720

A 6-letter permutation is selected at random from the letters NIMBLE: How many permutations are possible?

6! == 720

The menu at Valero's lists 7 kinds of salad, 11 entrees, and 9 kinds of desserts. How many different salad-entree-dessert- meals could you select?

7 times 11 times 9 = 693 meals

Express as ratio of factorials: 7*6*5*4

7! / 3!

Divide: 72! / 71!

72

Given 5 different physics book, 4 different math books, and 7 different english books: How many ways to arrange the physics and math books on a shelf if each subject is to remain together?

KEY WORD: Remain (4!) (5!) Times 2 === 5760 M P

How many ways can you form a 3 term expression of DISSIMILAR terms if you must choose the terms from the 9: 6x +3x-10x+x+5y-2y+7-8+2

____ _____ ______ x:4 y:2 4 times 2 times 3 = 24 ways #:3

In a state lottery, if you pick the correct sequence of 5 integers from 1 to 25, you win. How many ways can you pick a sequence of five different integers from 1 to 25?

____ ______ _____ ____ ______ 25 times 24 times 23 times 22 times 21 ==> 6375000

E, G, J, K, and B go to the movies. How many ways can they be seated in 5 consecutive seats if G must sit in between K and B?

_____ _____ ___ ___ ____ 2(__ ___ ___) 2 1 = 4 2 1 ( ___ _____ ____)2= 4 1 (_____ ______ ______)2 2= 4 ------ 12

How many ways can you arrange: 6 books from a set of 8 , on a shelf?

_____ _____ _____ _____ ____ _____ 8 7 6 5 4 3 ====> 20160

How many positive odd integers less than 1,000 can be formed by using digits 3,4,5,6?

_____ _____ _____ and _____ ____ and _____ 4 4 2 4 2 2 32+8+2= 42

How many even whole numbers between 3,000 and 30,000 can be made using only the digits 1,2,3,4,5 if repetition is allowed?

_____ _____ ______ _____ and _____ ____ ____ ______ ____ *3 5 5 2 + *2 5 5 5 2 150 + 500 = 650 * 3 b/c can't have anything lower than 3,000 , so can only have 3,4,5 for first term *2 b/c can't have anything greater than 30,000 , so can only have 1 or 2 for first term

How many 4 letter words can you make from the Hawaiian alphabet (12 letters) if each word must contain 4 different letters?

_____ _____ _______ ______ 12 11 10 9 === 11880

Kerry's Restaurant offers an early bird special before 5 with a choice of 5 entrees,soup or salad, potato or rice, one of three drinks and chocolate cakes. How many different dinner combos possible?

_____ ______ ______ ______ _______ 5 times 2 times 2 times 3 times 1 ==> 60 combos entrees soup/ potato/ drinks cake salad rice

Using the digits 2,4,5,8,9 how many 4 digit numbers can be formed if: Digits can be repeated?

_____ ______ ______ _______ 5 5 5 5 ====>625

How many ways are there to form a 3 letter sequence using the letters c,h,a,r,g,e ? Without repetition?

_____ ______ _______ 6 5 4 =====120

How many ways are there to form a 3 letter sequence using the letters c,h,a,r,g,e ? With repetition allowed?

_____ ______ _______ 6 letters in total 6 6 6 ===> 216

4 cards are picked from a 52 card deck and arranged on a table from left to right as drawn. How many ways can you make this 4 card arrangement?

_____ ________ _______ ________ 52 51 50 49 =====> 6497400

How many positive odd integers less than 100 can be formed by using digits 3,4,5,6?

______ _____ and ______ 4 2 + 2 ===>10

How many odd whole numbers less than 100 can be made using 1,2,4,7,9 , if repetition of digits is allowed?

______ _____ and ________ (5 * 3) + 3 ===> 18

Given 5 different physics book, 4 different math books, and 7 different english books: How many ways to select three books , one of each subject?

______ _______ ______ 4 5 7 ====> 140

A flight flying out of Chicago Airport has flights in London, Paris, Hawaii, Mexico City, Rome, and Tokyo. There are six pilots available. B/C of bad weather , flights to Paris and Rome are cancelled. Now there are 6 pilots available to fly to 4 cities. How many ways can the schedule be make if requests are not considered?

______ ________ ________ ______ 6 times 5 times 4 times 3 times ===> 360 ways

Using the digits 2,4,5,8,9 how many 4 digit numbers can be formed if: Each digit can only be used once?

_______ _______ ______ _______ 5 4 3 2 ====> 120

How many even three digit numbers can be formed using 1,2,4,6,7 if each digit can be used no more than once.

_______ _______ _______ 4 2 3 ===> 24

How many ways are there to form a 3 letter sequence using the letters c,h,a,r,g,e ? Without repetition, but must use g?

_______ _______ _______ 5 4 3 * five letters possible with G

How many 4 digit numbers are there if there are no leading zeros?

_______ ________ _________ ________ *9 10 10 10 ======> 9,000 * 9 bc can't have 0 in first term

Five chairs are arranged in a row. How many ways can 5 ppl be seated if twins Bill and Jill must sit next to each other?

________ _______ ________ _______ ________ ( 2 ) 3 2 1 ==> 12 ______ ________ _______ ______ _______ 3 ( 2 ) 2 1 ==> 12 _______ ________ _______ _______ ________ 3 2 ( 2 ) 1 ====> 12 _____ ________ _______ ______ ______ 3 2 1 ( 2 ) ====>12 12 times 4 = 48

The parking lot at Gypsum Bank has three spaces labeled "VP". However, there are seven "VP'S". In how many ways could these spaces be occupied by the VP's cars?

_________ _________ _______ 7 6 5 ==== 210

Suppose the prestige license plates are made using exactly 4 of the 26 letters in the alphabet. How many different prestige plates could be made if all 4 letters are different?

__________ __________ ___________ __________ 26 25 24 23 === 358800

How many ways are there to form a 3 letter sequence using the letters c,h,a,r,g,e ? With repetition and g is used at least once?

___________ ___________ __________ === 93 *3 ( 1 5 5) 75 *3( 1 1 5) 15 *3( 1 1 1) 3 * 3 : If I put G here... the other two numbers have to be 5 b/c they cannot be g

(n+1)! / n!

n+1

How many 4 digit numbers contain exactly one 8?

x 9 9 9 8 9 x 9 8 x 9 9 8 9 9 x 729+ 648+648+648= 2673