1. Counting Unit Review Sheet

2. 1. There are five choices of ice cream AND three choices of cookies. a)How many different desserts are there if you have one scoop of ice cream AND one cookie? _________ ________ Ice cream cookie 5 3____ Ice cream cookie 15 b) How many different desserts are there if you have either one scoop of ice cream OR a cookie? 5(ice cream) + 3(cookie) 8

3. 2. How many different 3-letter “words” can be formed from the letters in the word CANOE? ____ 5 4 3___ 60

4. 3. How many different ways can 5 children arrange themselves for a game of ring-around-the- rosie? (5 – 1)! 4! 4 3 2 1 = 24

5. 4. How many different ways can a teacher choose 10 homework problems from a set of 25? 25 C 10 25! = (25-10)! 10! 2524232221201918171615! 15! 10987654321 3,268,760

6. 5. How many different arrangements are there of the digits 166555? 6! = 2! 3! 6 5 4 3! 2 3! 60

7. 6. A child has 10 identically shaped blocks – 4 red, 3 green, 2 yellow, and 1 blue. How many different stacks of all 10 blocks are possible? 10! = 4! 3! 2! 1! 10 9 8 7 6 5 4! 4! 3 2 2 151,200 12 12,600

8. 7. How many ways can 10 people be seated around a circular table if the host and hostess cannot be seated together? (10 – 1)! = 362,880 If the host and hostess do sit together, they would be counted as one, so now it would be asking for 9 people seated in a circle. (9 – 1)! = 8! = 40320 So to find the ways they do not sit together, subtract the two answers 9! – 8! = 362,880 – 40320 = 322,560

9. 8. A committee of 4 is to be chosen from a club with 10 male and 12 female members. If at least 2 women must be chosen how many ways can this be done ____ ___ female male 12 C 2 10 C 2 12 C 2 10 C 2 + 12 C 3 10 C 1 + 12 C 4 10 C 0

10. 13. Find the number of arrangements of the word LEVELED 7! = 3! 2! 7 6 5 4 3! 3! 2 840 2 420

11. 14. How many 4 digit numbers can be made using the digits 0, 1, 2, 3, 4, 5 if repetition is not allowed? _ _ _ _ = 5 5 4 3 = 300

12. (How many 4 digit numbers can be made using the digits 0, 1, 2, 3, 4, 5 if repetition is not allowed?) 15. How many of them are odd? _ _ _ _ = 4 4 3 3 144

13. (How many 4 digit numbers can be made using the digits 0, 1, 2, 3, 4, 5 if repetition is not allowed?) 16. Do #14 if repetition is allowed _ _ _ _ = 5 6 6 6 1080

14. 17. How many ways can you answer a 15-question always- sometimes-never geometry quiz __ 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 14348907

15. 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

16. 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

17. 5. Fifteen people line up for concert tickets. a)How many different arrangements are possible? __________________ __________ _= 151413121110987654321 = 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? 1 ________________ __________ 1 = 11312111098765432 11 = 6,227,020,800

18. 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

19. 6) Using the letters A, B, C, D, E, F a)How many “words” can be made using all 6 letters? 6 P 6 = 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? 6 P 4 = 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? 6 P 2 + 6 P 3 = 65 + 654 = 30 + 120 = 150 g) If no repetition is allowed, how many words containing at least 5 letters can be made 6 P 5 + 6 P 6 = 720 + 720 = 1440

20. 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! 2! 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

21. 8. Suppose you have 3 red flags, 5 green flags, 2 yellow flags, and 1 white flag. Using all the flags in a row, how many distinguishable signals can be sent? 11! = 3! 5! 2!1! 11 10 9 8 7 6 5! = 3 2 5! 2 332,640 = 12 27,720

22. 9. How many ways can 7 people be seated in a circle? (7-1)! = 720

23. 10. If you have a dozen different flowers and wish to arrange them so there is one in the center and the rest in a circle around them, how many arrangements are possible? 12 (11-1)! = Center Circle 12 3,628,800 = 43,545,600

24. 11. Note: zero can never be the first digit of a “__-digit number”. a)How many 4- digit numbers contain no nines? __ __ 8 9 9 9 = 5832 b) How many 4- digit numbers contain AT LEAST ONE nine? __ __ 9 10 10 10 – 8 9 9 9 = 9000 – 5832 = 3168

25. 12. How many 10-letter words can you make if no letter can be repeated? Set up using the fundamental counting principle. __ __ __ __ __ __ __ 26252423222120 191817 = 1,927,522,397,000 Then using permutation notation 26 P 10 = 26! = (26 – 10)! 26! 16! 2625242322212019181716! 16!

26. 13. How many 26-letter words can be made if no repetition of a letter is allowed? 26!

27. 14) How ways can your homeroom (of 23 people) choose an ASC rep and a ASC alternate? 23 P 2 = 23 22 = 506

28. 15) Suppose we just want to select 2 people in the homeroom to serve on the ASC committee. How many 2-person groups are possible 23 C 2 = 23! = 21! 2! 23 22 = 2 253

29. 16) How many 5-card “hands” are possible when dealt from a deck of 52 cards? 52 C 5 = 52! = 47! 5! 52 51 50 49 48 47! = 47! 5 4 3 2 1 2,598,960

30. 17. Eight points are located on the circumference of a circle. You want to draw a triangle whose vertices are each one of these points. How many triangles are possible? _______ Starting Circle Vertex ___7!____ ___6!____ 5040 720 3,628,800

31. 18) Out of a class of 6 seniors and 5 juniors. I need to select a dance committee that must contain 2 seniors and 1 junior. How many different ways can this be done? 6 C 2 5 C 1 = 6! 5! = 4! 2! 4! 1! 6 5 4! 5 4! = 4! 2 4! 75