1. 11/30/2015MATH 106, Section 61 Section 6 More Ordered Arrangements Questions about homework? Submit homework!

2. 11/30/2015MATH 106, Section 62 Consider different arrangements of the word MARGIN. How many ways can the letters be arranged? How many ways can the letters be arranged keeping the two vowels together? How many ways can the letters be arranged keeping the two vowels separated by at least one consonant? P(6, 6) = 6! = 720 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 We can solve this by using … #1

3. 11/30/2015MATH 106, Section 63 We can solve this by using … the GOOD = ALL – BAD approach How many ways can the letters be arranged keeping the two vowels separated by at least one consonant? total number of possibilities total number of possibilities we are not interested in total number of possibilities we are interested in 720 – 240 =480

4. 11/30/2015MATH 106, Section 64 Consider different arrangements of the word CAULKING. How many ways can the letters be arranged? How many ways can the letters be arranged keeping the three vowels together? How many ways can the letters be arranged keeping at least two of the three vowels separated by at least one consonant? P(8, 8) = 8! = 40320 Arrange six slots for the five consonants and three vowels, and then Arrange the order for the three vowels. P(6, 6)  3! = 4320 40320 – 4320 =36000 #2

5. 11/30/2015MATH 106, Section 65 How many ways can the letters be arranged keeping the “A” and “U” together? How many ways can the letters be arranged keeping the “A” and “U” separated by at least one letter? How many ways can the letters be arranged to end in “ING”? Arrange seven slots, six for letters other than “A” and “U” and one for the “A” and “U”, and then Arrange the order for the “A” and “U”. P(7, 7)  2! = 10080 40320 – 10080 =30240 P(5, 5) = 120

6. 11/30/2015MATH 106, Section 66 How many ways can the letters be arranged to not end in “ING”? How many ways can the letters be arranged to have “ING” somewhere in the arrangement? 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 – 120 =40200 P(6, 6) = 720 40320 – 720 =39600

7. 11/30/2015MATH 106, Section 67 How many ways can 5 people be placed in an order to sit in a row? sit in a circle? A B CD E ABCDEABCDE P(5, 5) = 120 is the same as D E AB C Suppose we decide to always place A “on top”. We must now count the number of ways to seat the remaining 4 people. P(4, 4) = 24 #3

8. 11/30/2015MATH 106, Section 68 Observe that each circular arrangement corresponds to n straight line arrangements. Let’s generalize this to arrange n objects in a circle … 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

9. 11/30/2015MATH 106, Section 69 Six friends (Joey, Chandler, Ross, Monica, Rachel, and Phebee) are to be seated. How many seating arrangements are there at a round dinner table? How many seating arrangements are there at a round dinner table if Chandler and Monica must sit together? How many seating arrangements are there at a round dinner table if Ross and Rachel refuse to sit together? (6 – 1)! = 120 (5 – 1)!  2! = 48 120 – 48 = 72 #4

10. 11/30/2015MATH 106, Section 610 How many seating arrangements are there at a round dinner table if they sit boy/girl/boy/girl/boy/girl? 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? (3–1)!  3! = 12 (3–1)!  3  3! = 36 Arrange the boys in a circle, and then arrange the girls in between the boys. Arrange the boys in a circle, and then choose between which boys the girls sit, and then arrange the girls.

11. 11/30/2015MATH 106, Section 611 A room contains 8 people. How many ways can we select a group of 2 of these people? How many ways can we select a group of 6 of these people? How many ways can we select a group of 3 of these people? How many ways can we select a group of 5 of these people? C(8,2) = 28 C(8,6) = 28 C(8,3) = 56 #5 C(8,5) = 56

12. 11/30/2015MATH 106, Section 612 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.

13. 11/30/2015MATH 106, Section 613 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.