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Section 6 More Ordered Arrangements

Section 6 More Ordered Arrangements. Questions about homework? Submit homework!. #1. 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?

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Section 6 More Ordered Arrangements

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  1. Section 6More Ordered Arrangements • Questions about homework? • Submit homework! MATH 106, Section 6

  2. #1 • 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 … MATH 106, Section 6

  3. 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 total number of possibilities we are not interested in – 240 = 480 720 MATH 106, Section 6

  4. #2 • 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 – 4320 = 36000 40320 MATH 106, Section 6

  5. 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 – 10080 = 30240 40320 P(5, 5) = 120 MATH 106, Section 6

  6. 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? – 120 = 40200 40320 P(6, 6) = 720 – 720 = 39600 40320 MATH 106, Section 6

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

  8. 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 MATH 106, Section 6

  9. #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? • 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 MATH 106, Section 6

  10. 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? Arrange the boys in a circle, and then arrange the girls in between the boys. (3–1)!  3! = 12 Arrange the boys in a circle, and then choose between which boys the girls sit, and thenarrange the girls. (3–1)!  3 3! = 36 MATH 106, Section 6

  11. #5 • 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 C(8,5) = 56 MATH 106, Section 6

  12. 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. MATH 106, Section 6

  13. 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. MATH 106, Section 6

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