Permutations and Combinations Smart Notes.notebook
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• Discrete Math is concerned with counting.
Permutations and Combinations
http://ed.ted.com/lessons/howmanywayscanyouarrangeadeckofcardsyannaykhaikin#watch
https://docs.google.com/a/unionacademy.org/forms/d/1oijns3g7lr5oGFkja1GeUjffm5eIa8ka33Uc3L6Y3Y/viewform
Ted TV:How many ways can you arrange a deck of cards? Yannay Khaikin
Google Form: Permutations and Combinations Quick Quiz #1
• Multiplication Principle of Counting Identify how many options are possible for each spot and then multiply them.
– The number of ways a pattern can be written is the number of terms “factorial”.
Example: How many ways can the letters "ABC" be rearranged?
Answer: 3 terms = 3 ! = 3 x 2 x 1 = 6 possible patterns
Question 1: How many ways can the letters ABCDE be rearranged?
Answer: 5! = 5 x 4 x 3 x 2 x 1 = 120 possible patterns
Foldable
Permutations and Combinations
Permutations and Combinations Smart Notes.notebook
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Permutations and CombinationsFoldable Example 1:
Permutations and CombinationsFoldable Example 2:
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Example: How many ways can you have a license plate can have 3 letters and 3 numbers?
» Assume that it is OK to have repeating letters and numbers
Answer: Letters: There are 26 letters in the alphabet and 3 places to put letters so: 26 x 26 x 26 Digits: There are 10 digits and 3 places to put numbers so: 10 x 10 x 10
Multiply it all together: 26 x 26 x 26 x 10 x 10 x 10 = 17576000 combinations
Question 2: How many ways can you have a license plate can have 4 letters and 3 numbers (with repeating)?
Answer: 26 x 26 x 26 x 26 x 10 x 10 x 10 = 456976000
Question 3: How many ways can you have a license plate can have 3 letters and 4 numbers (with repeating)?
Answer: 26 x 26 x 26 x 10 x 10 x 10 x 10 = 175760000
Permutations and Combinations
Example: How many ways can you have a license plate can have 3 letters and 3 numbers?
» Assume that it is NOT OK to have repeating letters and numbers
Answer: Letters: There are 26 letters in the alphabet and 3 places to put letters so: 26 x 25 x 24 Digits: There are 10 digits and 3 places to put numbers so: 10 x 9 x 8
Multiply it all together: 26 x 25 x 24 x 10 x 9 x 8 = 11232000 combinations
Question 4: How many ways can you have a license plate can have 4 letters and 3 numbers without repeating?
Answer: 26 x 25 x 24 x 23 x 10 x 9 x 8 = 258336000
Question 5: How many ways can you have a license plate can have 3 letters and 4 numbers without repeating?
Answer: 26 x 25 x 24 x 10 x 9 x 8 x 7 = 78624000
Permutations and Combinations
Permutations and Combinations Smart Notes.notebook
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Example: The 8th grade class has 9 girls and 15 boys. How many different girlboy dates are possible?
Answer: How many girls? = 9 girls
How many boys? = 15 boys
Multiplication Rule says to multiply the options = 9 x 15 = 135 different date combinations
Question 6: How many different dates can be made if there are 14 girls and 15 boys?
Answer: 14 x 15 = 210
Permutations and Combinations
• permutation: the pattern or order of the objects THE ORDER MATTERS!
Permutations and Combinations
Foldable Example 3:
Permutations and Combinations Smart Notes.notebook
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Question 7: Baby Albert wants to arrange 7 blocks in a row. How many different ways can this
be done?
• TI TIP: to easily calculate 7! on a regular blank screen, type "7", MATH, PRB, #4
Answer: 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
Permutations and Combinations
Question 8: How many different permutations are there with the letters in the word "PENCILS" if all the letters are used without repetition?
Question 9: How many different permutations are there in the name HAGLER if all the letters are used without repetition?
Question 10: How many different permutations are there in the name HAGLER if all the letters are used WITH repetition?
Permutations and Combinations
Answer: 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
Answer: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
Answer: 6 x 6 x 6 x 6 x 6 x 6 = 46656
Permutations and Combinations Smart Notes.notebook
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Distinguishable Permutations
There are 10! distinguishable permutations of a 10set containing 10 distinguishable objects.
If an 10set contains 2 objects of the same kind, and 8 objects of different kinds, then the number of distinguishable permutations is: 10! = 1814400 unique, distinguishable permutations
2!
Question 11: How many distinguishable 6 letter words can be formed using the letters in the word "HAWAII"?
Permutations and Combinations
Answer: 6! / 2!2! = 180
Question 12: How many distinguishable words can be formed using ALL the letters in the name BRIANNA?
Question 13: How many distinguishable words can be formed using ALL the letters in the word MISSISSIPPI?
Permutations and Combinations
Answer: 7! / 2!2! = 1260
Answer: 11! / 4!4!2! = 34650
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Permutations and CombinationsFoldable Example 5:
What if you don't want to use ALL the letter choices you just need to pick a couple of them?
Example: How many 3 letter words can be made (without repeating)?
Answer: There are 26 letters in the alphabet. You want to pick 3 of them.
26 x 25 x 24 = 15600 three lettered words (without repeating)
TI TIP: on a blank screen, type 26, MATH, PRB, nPr , 3, ENTER = 15600
This is a 26P3 = 15600
Question 14: How many 4 letter words can be made from the alphabet? (without repetition)
Permutations and Combinations
Answer: 26 P 4 = 358800
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Permutations and CombinationsFoldable Example 4:
Quiz Time
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Combinations of n objects taken r at a time: when you are interested in the WAYS to select the r objects, but NOT the specific order they are arranged.
Permutations and Combinations
Example: How many WAYS can you pick 3 books from a bookshelf with 12 books on it?
Answer: This is a "combination" situation because the "order" that we pick the books doesn't matter. Perform a 12C3 = 220
TI TIP: on a blank screen, type 12, MATH, PRB, nCr , 3, ENTER = 220
This is a 12C3 = 220
Question 15: How many ways can you pick 3 students to be on a committee out of a class of 14?
Permutations and Combinations
Answer: 14 C 3 = 364
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Permutations and CombinationsFoldable Example 1:
Permutations and CombinationsFoldable Example 2:
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Permutations and CombinationsFoldable Example 3:
Question 16: Identify this as a Permutation or Combination.
(a) Detective Casey will read the files on four unsolved cases from a list of fourteen. She does not care what order they are read in. Combination
(b) How many 6 person committees can be made out of a class of 25 students? Combination
(c) How many ways can 11 racers win the top 3 positions in the race? Permutation
Question 17: Now answer the above scenarios:
(a) 1001
(b) 177100
(c) 990
Permutations and Combinations
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Combinations can also be written: ( ) = 8C5 = 5685
Permutations and Combinations
Quiz Time(and then TEST TIME)