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

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 Engineering
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
𝑛! 𝑎!𝑏!𝑐!…

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
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 purchasing three 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
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

17. 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

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

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

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

22. Answer

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

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

25. 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

26. Answer
= 20x10x26x10x26x10 = 13,520,000
20 choices for the first letter ( 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.

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

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

29. 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

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

31. 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

32. Answer
C(11,3) = 11! 8! 3! = 165

33. 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

34. Answer
C(52,5) = 52! 47! 5! =2,598,960

35. The game of euchre uses only 24 cards from a standard deck
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

36. Answer
C(24,5) = 24! 19! 5! = 42,504

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

38. Answer

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

40. Answer
= 3! x 3! + 3! x 3!
3 𝐵 𝑥 3 𝐺 𝑥 2 𝐵 𝑥 2 𝐺 𝑥 1 𝐵 𝑥 1 𝐺 + 3 𝐺 𝑥 3 𝐵 𝑥 2 𝐺 𝑥 2 𝐵 𝑥 1 𝐺 𝑥 1 𝐵
= 72

41. Laura has ‘lost’ Jordan’s phone number
Laura has ‘lost’ Jordan’s phone number. All she can remember is that it did not contain a 0 or 1 in the first three digits. How many 7 digit #’s are possible

42. Answer
= 8 x 8 x 8 x 10 x 10 x 10 x 10
= 5,120,000