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Permutations and Combinations

MDM 4U: Mathematics of Data Management Unit: Counting and Probability By: Mr. Allison and Mr. Panchbhaya. Permutations and Combinations. Specific Expectations. Strand 2.1

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Permutations and Combinations

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  1. MDM 4U: Mathematics of Data Management Unit: Counting and Probability By: Mr. Allison and Mr. Panchbhaya Permutations and Combinations

  2. Specific Expectations • Strand 2.1 • Recognize the use of permutations and combinations as counting techniques with advantages over other counting techniques • Strand 2.2 • Solve simple problems using techniques for counting permutations and combinations, where all objects are distinct Learning Goals • Make connections between, and learn to calculate various permutations and combinations • Learn to behave in class

  3. Agenda of the Day • Probability Video • Review • Worksheet • Game show Activity

  4. How many combinations would it take for the tire to attach itself back to the car?

  5. Real Life Examples • Video game designers • to assign appropriate scoring values • Engineering • new products tested rigorously to determine how well they work • Allotting numbers for: • Credit card numbers • Cell phone numbers • Car plate numbers • Lottery

  6. Factorials • The product of all positive integers less than equal or equal to n n! = n x (n – 1) x (n – 2) x … x 2 x 1 5! =5 x 4 x 3 x 2 x 1 = 120

  7. Permutations • Ordered arrangement of objects selected from a set • Ordered arrangement containing a identical objects of one kind is

  8. Combinations • Collection of chosen objects for which order does not matter

  9. Speed Round: The sports apparel store at the mall is having a sale. Each customer may choose exactly two items from the list, and purchase them both. The trick is that each 2-item special must have two different items (for example, they may not purchase two T-shirts at the same time). What are all the different combinations that can be made by choosing exactly two items?

  10. 15 combinations are possible

  11. Q – How many combinations are made if you were purchasingthree items instead of two?

  12. 1. A club of 15 members choose a president, a secretary, and a treasurer in • 455 ways • 6 ways • 2730 ways

  13. 2. The number of debate teams formed of 6 students out of 10 is: • 151200 • 210 • 720

  14. 3. A student has to answer 6 questions out of 12 in an exam. The first two questions are obligatory. The student has: • 5040 • 210 • 720

  15. 4. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done. • 564 • 645 • 735 • 756 • None of the above

  16. 5. In how many different ways can the letters of the word “LEADING” be arranged in such a way that the vowels • 360 • 480 • 720 • 5040 • None of the above

  17. 6. How many permutations of 4 different letters are there, chosen from the twenty six letters of the alphabet (repetition is not allowed)?

  18. Answer The number of permutations of 4 digits chosen from 26 is 26P4 = 26 × 25 × 24 × 23 = 358,800

  19. How many paths are there to the top of the board?

  20. Answer

  21. How many 4 digit numbers can be made using 0-7 with no repeated digits allowed? • 5040 • 4536 • 2688 • 1470

  22. Answer • = 7x7x6x5 = 1470 • First digit of a number can not be ‘0’

  23. No postal code in Canada can begin with the letters D,F,I,O,Q,U, but repeated letters are allowed and any digit is allowed. How many postal codes are possible in Canada? • 11,657,890 • 13,520,000 • 14,280,000 • 12,240,000

  24. Answer • = 20x10x26x10x26x10 = 13,520,000 • 20 choices for the first letter (26 - 6 that cannot be chosen. 10 choices for the digit (0-9). • 26 choices for the 3 position (2nd letter) • then 10 choice for the 4th position • Then 26 and 10 since you can again repeat numbers and letters.

  25. Using digits 0 – 9, how many 4 digit numbers are evenly divisible by 5 with repeated digits allowed? • 1400 • 1600 • 1800 • 1500

  26. Answer • 9 × 10 × 10 × 2 = 1800 • First # can’t be ‘0’ • Last # has to be ‘5’ or ‘0’

  27. How many ways can you arrange the letters in the word REDCOATS if it must start with a vowel • 15,120 • 14,840 • 15,620 • 40,320

  28. Answer • 3* × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 15,120 • EOA are your 3 choices

  29. How many groups of 3 toys can a child choose to take on a vacation from a toy box containing 11 toys? • 990 • 1331 • 165 • 286

  30. Answer • C(11,3) 165

  31. If you have a standard deck of cards how many different hands exists of 5 cards • 2,598,960 • 3,819,816 • 270,725 • 311,875,200

  32. Answer • C(52,5)

  33. The game of euchre uses only 24 cards from a standard deck. How many different 5 card euchre hands are possible? • 7,962,624 • 42,504 • 5,100,480 • 98,280

  34. Answer • C(24,5) 42,504

  35. Solve for n 3(nP4) =n-1P5 • 8 • 10 • 2 • 5

  36. Answer

  37. How many ways can 3 girls and three boys sit in a row if boys and girls must alternate?

  38. Answer • = 3! x 3! + 3! x 3! • = 72

  39. Laura has ‘lost’ Jordan’s phone number. All she can remember is that it did not contain a0 or 1 in the first three digits. How many 7 digit #’s are possible

  40. Answer • = 8 x 8 x 8 x 10 x 10 x 10 x 10 • = 5,120,000

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