## Lecture Notes

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## Lecture Notes

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- Pages:
- 3
- School:
- Louisiana State University
- Course:
- Math 2020 - Solving Discrete Pro

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Math2020 Solutions to question 7 of final guide 1 a How many ways are there to select three different pieces of fruit from a bowl of 5 different fruits SOLUTION Ways to choose 3 from 5 is 5 5 4 3 10 1 2 3 3 b You go to the supermarket to buy three peices of fruit which may or may not all be the same The supermarket sells 5 different kinds of fruit How many different possibilities are there for what you come home with SOLUTION Ways to choose 3 from 5 types of things order doesn t matter repeats are allowed 3 5 1 7 7 6 5 35 1 2 3 5 1 3 c You are painting the windows frames on the front of a house There are three windows and 5 choices of colors How many different ways could you paint them if there is no requirement that they have to be different colors from each other SOLUTION 3 choices from 5 possible types Order does matter Repeats are allowed 5 5 5 125 d You are painting the windows frames on the front of a house There are three windows and 5 choices of colors How many different ways could you paint them if there is a requirement that they all have to be different colors from each other SOLUTION 3 choices from 5 possible types Order does matter Repeats are not allowed P 5 3 5 4 3 60 e You have 5 pieces of fruit which you are lining up on your kitchen counter Your fruits are 2 identical apples 2 identical bananas and 1 orange How many different ways can you line them up SOLUTION Ways of ordering 5 things of three types 2 of type 1 2 of type 2 1 of type 3 Order matters 5 5 4 3 2 1 30 2 2 1 2 2 1 f You have 5 pieces of fruit which you are lining up on your kitchen counter Your fruits are 2 identical apples and 3 identical bananas How many different ways can you line them up SOLUTION Ways of ordering 5 things of two types 2 of type 1 3 of type 2 Order matters 5 4 3 2 1 5 10 2 3 2 6 1 2 g You have 5 pieces of fruit of which you are lining up three on your kitchen counter you will eat the remaining 2 Your fruits are 2 identical apples 2 identical bananas and 1 orange How many different ways can you line three of them them up SOLUTION Choose 3 from 5 order does matter some objects are identical 2 of type 1 2 of type 2 1 of type 3 Suppose we lined up all five Then there are 5 30 2 2 1 ways as in question e We can eat the last two things in the line But when we do this we count some arrangements twice E g writing A for apple B for banana O for orange we have AABOB AABBO both give AAB when the last two fruits are removed Any arrangement with the last two fruits different will lead to an arrangement of three fruits being counted twice How many ways are there to arrange the fruits with the last two different If X is the number with the last two different and Y the number with the last two the same then X Y is the total number so X Y 30 and X 30 Y If the last two are the same they are either two apples or two bananas If we have two apples at the end there are two bananas and one orange left to arrange so there are 3 3 2 1 ways to arrange the remaining fruit If we have two bananas at the end there are similarly 3 3 2 1 ways to arrange the remaining fruit So there are 3 3 6 ways to arrange fruit with the last two the same So there are 30 6 24 ways to arrange with the last two different Each of these 24 ways gives a double count of the first three pieces of fruit So there are 12 ways to arrange the fruits with two different pieces of fruit left over There are 6 ways with the same fruit left So there are 6 12 18 ways to arrange three of the 5 fruits ANSWER 18 h You go to the cinema with three friends If you all sit in four adjacent seats how many different ways can you sit together SOLUTION Ways to order 4 different things P 4 4 4 24 3 i You go to the cinema with three friends Two of them must sit in adjacent seats to each other but noone else cares about seating How many ways can you sit together in four adjacent seats SOLUTION Ways to order 4 things but two must be adjacent There are 24 total ways without this requirement as in previous question If two people are always together we can think of them as one unit and just choose an ordering of three things P 3 3 3 6 But for each way of putting those two people together there are two ways since they can swap seats with each other and still be together So there are 2P 3 3 2 6 12 ways j You go to the cinema with three friends Two of them will not sit next to each other How many ways are there for you all to sit together in four adjacent seats SOLUTION There are 24 total ways as in question h There are 12 ways where these two people are together as in question i So this leaves 24 12 ways where they are not together ANSWER 12

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