Permutations and Combinations
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1 Smart Notes.notebook Discrete Math is concerned with counting. Ted TV:How many ways can you arrange a deck of cards? Yannay Khaikin many ways can you arrange a deck of cardsyannay khaikin#watch Google Form: Quick Quiz #1 1oijns3g7lr5oGFkja1GeUjffm5eIa8 ka33uc3l6y3y/viewform 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 1
2 Smart Notes.notebook Foldable Example 1: Foldable Example 2: 2
3 Smart Notes.notebook 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 = 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 = 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 = 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 = 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 = 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 =
4 Smart Notes.notebook Example: The 8th grade class has 9 girls and 15 boys. How many different girl boy 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 permutation: the pattern or order of the objects THE ORDER MATTERS! Foldable Example 3: 4
5 Smart Notes.notebook Question 7: Baby Albert wants to arrange 7 blocks in a row. How many different ways can this be done? Answer: 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 TI TIP: to easily calculate 7! on a regular blank screen, type "7", MATH, PRB, #4 Question 8: How many different permutations are there with the letters in the word "PENCILS" if all the letters are used without repetition? Answer: 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 Question 9: How many different permutations are there in the name HAGLER if all the letters are used without repetition? Answer: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 Question 10: How many different permutations are there in the name HAGLER if all the letters are used WITH repetition? Answer: 6 x 6 x 6 x 6 x 6 x 6 =
6 Smart Notes.notebook Distinguishable Permutations There are 10! distinguishable permutations of a 10 set containing 10 distinguishable objects. If an 10 set contains 2 objects of the same kind, and 8 objects of different kinds, then the number of distinguishable permutations is: 10! = unique, distinguishable permutations 2! Question 11: How many distinguishable 6 letter words can be formed using the letters in the word "HAWAII"? Answer: 6! / 2!2! = 180 Question 12: How many distinguishable words can be formed using ALL the letters in the name BRIANNA? Answer: 7! / 2!2! = 1260 Question 13: How many distinguishable words can be formed using ALL the letters in the word MISSISSIPPI? Answer: 11! / 4!4!2! =
7 Smart Notes.notebook Foldable 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 = three lettered words (without repeating) TI TIP: on a blank screen, type 26, MATH, PRB, n P r, 3, ENTER = This is a 26 P 3 = Question 14: How many 4 letter words can be made from the alphabet? (without repetition) Answer: 26 P 4 =
8 Smart Notes.notebook Foldable Example 4: Quiz Time 8
9 Smart Notes.notebook 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. 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 12 C 3 = 220 TI TIP: on a blank screen, type 12, MATH, PRB, n C r, 3, ENTER = 220 This is a 12 C 3 = 220 Question 15: How many ways can you pick 3 students to be on a committee out of a class of 14? Answer: 14 C 3 = 364 9
10 Smart Notes.notebook Foldable Example 1: Foldable Example 2: 10
11 Smart Notes.notebook Foldable 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) (c)
12 Smart Notes.notebook 8 Combinations can also be written: ( ) 5 = 8 C 5 = 56 Quiz Time (and then TEST TIME) 12
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