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Counting Unit

Counting Unit. Review Sheet. 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.

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Counting Unit

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  1. Counting Unit Review Sheet

  2. 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 manydifferent desserts are there if you have either one scoop of ice cream OR a cookie? 5(ice cream) +3(cookie) 8 1. There are five choices of ice cream AND three choices of cookies.

  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? 25C10 25! = (25-10)! 10! 25•24•23•22•21•20•19•18•17•16•15! 15! 10•9•8•7•6•5•4•3•2•1 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 C2•10C2 12 C2 10C2 + 12C3 10 C1 +12C4 10 C0

  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 ifrepetition 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. How many different routes can you take for the trip to Philadelphia by way of Trenton? ________ • _________ Trenton Philadelphia ___4____• ___3_____ 12 3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia.

  16. How many outfits can you have consisting of a shirt, a pair of pants, and a jacket? ______•______•______ Shirts Pants Jackets ___6__•__10__•__3___ 180 4. You have 10 pairs of pants, 6 shirts, and 3 jackets.

  17. How many different arrangements are possible? __•__•__•__•__•__•__•__•__ •__•__•__•__•__• _= 15•14•13•12•11•10•9•8•7•6•5•4•3•2•1 = 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 = 1•13•12•11•10•9•8•7•6•5•4•3•2•1•1 = 6,227,020,800 5. Fifteen people line up for concert tickets.

  18. How many “words” can be made using all 6 letters? 6• 5 • 4 • 3 • 2 • 1 = 720 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•5•4•3 = 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•5+6•5•4 = 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 6) Using the letters A, B, C, D, E, F

  19. How many “words” can be made using all 6 letters? 6P6 = 6• 5 • 4 • 3 • 2 • 1 = 720 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? 6P4 = 6•5•4•3 = 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? 6P2 + 6P3 = 6•5 + 6•5•4 = 30 + 120 = 150 g) If no repetition is allowed, how many words containing at least 5 letters can be made 6P5 + 6P6 = 720 + 720 = 1440 6) Using the letters A, B, C, D, E, F

  20. GREAT __•__•__•__•__ 5 • 4 • 3 • 2 • 1 5! 120 FOOD 4! 2! 4 • 3 • 2! 2! 12 TENNESSEE 9!_________ 4! 2! 2!1! 9 • 8 • 7 • 6 • 5 • 4! 4! 2 • 2 15,120 4 3,780 7. How many distinguishable permutations can be made using all the letters of:

  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. 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 11. Note: zero can never be the first digit of a “__-digit number”.

  25. Set up using the fundamental counting principle. __ • __ • __ • __ • __ •__ • __ • __ • __ • __ 26•25•24•23•22•21•20• 19•18•17 = 1,927,522,397,000 Then using permutation notation 26 P10 = 26! = (26 – 10)! 26! 16! 26•25•24•23•22•21•20•19•18•17•16! 16! 12. How many 10-letter words can you make if no letter can be repeated?

  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 P2 = 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 C2 = 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 C5 = 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 C2 • 5 C1 = 6! • 5! = 4! 2! 4! 1! 6 • 5 •4! • 5 • 4! = 4! 2 4! 75

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