Section 6 More Ordered Arrangements Questions about homework

Section 6 More Ordered Arrangements Questions about homework? Submit homework! 12/7/2020 MATH 106, Section 6 1

#1 Consider different arrangements of the word MARGIN. How many ways can the letters be arranged? P(6, 6) = 6! = 720 How many ways can the letters be arranged keeping the two vowels together? Arrange five slots for the four consonants and the pair of vowels, and then Arrange the order for the two vowels. P(5, 5) 2! = 240 How many ways can the letters be arranged keeping the two vowels separated by at least one consonant? We can solve this by using … 12/7/2020 MATH 106, Section 6 2

How many ways can the letters be arranged keeping the two vowels separated by at least one consonant? We can solve this by using … the GOOD = ALL – BAD approach total number of possibilities we are interested in total number of possibilities we are not interested in 720 – 240 = 480 12/7/2020 MATH 106, Section 6 3

#2 Consider different arrangements of the word CAULKING. How many ways can the letters be arranged? P(8, 8) = 8! = 40320 How many ways can the letters be arranged keeping the three vowels together? Arrange six slots for the five consonants and three vowels, and then Arrange the order for the three vowels. P(6, 6) 3! = 4320 How many ways can the letters be arranged keeping at least two of the three vowels separated by at least one consonant? 40320 – 4320 = 36000 12/7/2020 MATH 106, Section 6 4

How many ways can the letters be arranged keeping the “A” and “U” together? Arrange seven slots, six for letters other than “A” and “U” and one for the “A” and “U”, P(7, 7) 2! = 10080 and then Arrange the order for the “A” and “U”. How many ways can the letters be arranged keeping the “A” and “U” separated by at least one letter? 40320 – 10080 = 30240 How many ways can the letters be arranged to end in “ING”? P(5, 5) = 120 12/7/2020 MATH 106, Section 6 5

How many ways can the letters be arranged to not end in “ING”? 40320 – 120 = 40200 How many ways can the letters be arranged to have “ING” somewhere in the arrangement? P(6, 6) = 720 How many ways can the letters be arranged to have “ING” nowhere in the arrangement, even though each of the letters I, N, and G are included in the arrangement? 40320 – 720 = 39600 12/7/2020 MATH 106, Section 6 6

#3 How many ways can 5 people be placed in an order to sit in a row? A B C D E P(5, 5) = 120 sit in a circle? Suppose we decide to always place A “on top”. We must now count the number of ways to seat the remaining 4 people. A E D B D C is the same as C E B A P(4, 4) = 24 12/7/2020 MATH 106, Section 6 7

Let’s generalize this to arrange n objects in a circle … • Observe that each circular arrangement corresponds to n straight line arrangements. • Let c be the number of circular arrangements of n objects, then there are c n straight line arrangements. • We know the number of straight line arrangements is P(n, n) = n! • Using a combinatorial proof, we have 12/7/2020 MATH 106, Section 6 8

#4 Six friends (Joey, Chandler, Ross, Monica, Rachel, and Phebee) are to be seated. How many seating arrangements are there at a round dinner table? (6 – 1)! = 120 How many seating arrangements are there at a round dinner table if Chandler and Monica must sit together? (5 – 1)! 2! = 48 How many seating arrangements are there at a round dinner table if Ross and Rachel refuse to sit together? 120 – 48 = 72 12/7/2020 MATH 106, Section 6 9

How many seating arrangements are there at a round dinner table if they sit boy/girl/boy/girl? Arrange the boys in a circle, and then arrange the girls in between the boys. (3– 1)! 3! = 12 How many seating arrangements are there at a round dinner table if all of the boys sit together and all of the girls sit together? Arrange the boys in a circle, and then choose between which boys the girls sit, and then arrange the girls. (3– 1)! 3 3! = 36 12/7/2020 MATH 106, Section 6 10

#5 A room contains 8 people. How many ways can we select a group of 2 of these people? C(8, 2) = 28 How many ways can we select a group of 6 of these people? C(8, 6) = 28 How many ways can we select a group of 3 of these people? C(8, 3) = 56 How many ways can we select a group of 5 of these people? C(8, 5) = 56 12/7/2020 MATH 106, Section 6 11

Homework Hints: In Section 6 Homework Problem #5 notice that sitting six people in seven seats is essentially the same as thinking of the empty seat as being for a seventh person and sitting seven people in seven seats. 12/7/2020 MATH 106, Section 6 12

Quiz #1 NEXT CLASS! Be sure to do the review problems for this, quiz posted on the internet. The link can be found in the course schedule. 12/7/2020 MATH 106, Section 6 13