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

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

Agenda of the Day Probability Video Review Worksheet Game show Activity

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

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

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

Permutations Ordered arrangement of objects selected from a set Ordered arrangement containing a identical objects of one kind is 𝑛! 𝑎!𝑏!𝑐!…

Combinations Collection of chosen objects for which order does not matter

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?

15 combinations are possible

Q – How many combinations are made if you were purchasing three items instead of two?

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

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

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

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

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

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

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

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

Answer

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

Solve for n 3(nP4) =n-1P5 8 10 2 5

Answer

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

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

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

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