1. 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

2. 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

3. 5. Fifteen people line up for concert tickets. a)How many different arrangements are possible? 15141312111098 7654321 = 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? 11312111098 76543211 = 6,227,020,800

4. 6) Using the letters A, B, C, D, E, F a)How many “words” can be made using all 6 letters? 6 5 4 3 2 1 = 720 b)How many of these words begin with E ? 1 5 4 3 2 1 = 120 c) How many of these words do NOT begin with E? 720 –120 = 600 d) How many 4-letter words can be made if no repetition is allowed? 6543 = 360 e) How many 3-letter words can be made if repetition is allowed? 6 6 6 = 216 f) How many 2 OR 3 letter words can be made if repetition is not allowed? 65+654 = 30 + 120 = 150 g) If no repetition is allowed, how many words containing at least 5 letters can be made? (both letter 6a) 720 + 720 = 1440

5. 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

6. 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 1 1680 6 280

7. EX:  Find the value of 6! _ 4! x 2! Short Method 6 5 4! 4! 2 1 30 2 15

8. EX:  Find the value of 12! _ 3! x 9! Short Method 12 11 10 9! 3 2 1 9! 1320 6 220

9. 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. PopPpooPpopPpoPpPo

10. 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

11. The number of distinguishable permutations of the 5 letters in daddy is 5! 3! since the three d’s are alike and can be permuted in 3! ways.

12. DEF:  Number of Distinguishable Permutations Given n objects in which a of them are alike, the number of distinguishable permutations of the n objects is n! a!

13. 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

14. 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

15. 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 1 332640 12 27720

16. 16.4 Circular Permutations OBJ:  To find the number of possible permutations of objects in a circle

17. 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 position ABC ACB BAC BCA CAB CBA

18. 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.

19. 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

20. 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

21. DEF:  Number of Circular Permutations The number of circular permutations of n distinct objects is (n-1)!

22. 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

23. 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! 12 c)TENNESSEE 9! 4! 2! 2!1! 9 8 7 6 5 4! 4! 2 2 15,120 4 = 3,780