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Alice wants to make a fruit salad using 10 pieces of fruit. She can buy three types of fruits at the supermarket: Apples (A), Bananas (B), and Cranberries (C). She can take as many of each type of fruit as she wants but she can only have 10 in the end. One possible fruit salad can be made of 5 apples, 3 Bananas, and 2 Cranberries.

 

(a) How many types of fruit salads can Alice make?

 

(b) Given that she wants at least two types of fruits in the salad, how many types of fruit salads can Alice make?

 Jan 20, 2020
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For these types of questions, there is a method called stars and bars that we can use for part a.

 

This method is as follows:

 

Let's put ten indistinguishable fruits in a line(the "stars")

 

F F F F F F F F F F

 

Now, to separate this line of fruits into apples, bananas, and cranberries, we can add "bars". 

 

F F F | F F F F F F F

^                ^              ^

apples     bananas     cranberries 

Since we have an unlimited number of each fruit, it is a matter of how many ways we can order 10 identical fruits and 2 identical bars.

 

\(salads = {12! \over 10!2!}\)

a) 66 

 

For the second part, you can bounce off of part a. The only salads that don't have two or more fruits are "plain", they only have one type of fruit in it. There are only 3 fruits, the three cases that don't work are, 10 apples, 10 bananas and 10 cranberries. Now, all we have to do is subtract 3 from the total amount.

 

66-3=63

 

b)63

(this is my first answer, I hope I'm correct!)

 Jan 20, 2020

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