SOLUTION: natasha is going to order an ice cream cone with three different toppings. she can choose from the following toppings; crushed nuts; chocolate syrup; stawberry syrup; chocolate chi

Question 113054: natasha is going to order an ice cream cone with three different toppings. she can choose from the following toppings; crushed nuts; chocolate syrup; stawberry syrup; chocolate chips; M-and-Ms. in how many ways is it possible to make the selection?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Actually, this question can be answered in a number of different ways. The problem doesn't say whether Natasha is required to select three DIFFERENT toppings. In other words, if she had a serious thing for strawberries, could she pick strawberry syrup twice counting as two of the three toppings? Also, the problem doesn't say whether the order in which the toppings go on the cone make a difference? In other words, is crushed nuts THEN chocolate syrup THEN strawberry syrup a different (and countable) way to make the selection than chocolate syrup THEN crushed nuts THEN strawberry syrup. I'll show you how to solve the problem four different ways, and then you can ask your instructor what was meant.

Either way, we need to know the number of toppings available, so count them. I count 5.

: Three different toppings must be selected AND the order of their application makes a difference.
There are then 5 different ways to select the first topping. For each of those ways to pick the first topping, there are 4 different ways to pick the second topping (one less, because one of them is already 'used'), and after that, three ways to pick the final topping. So that leaves us with:

ways to select the toppings.

: Three different toppings must be selected BUT the order of their application does not make a difference.

Once the three toppings have been selected, there are 3 ways one of them could have been first, 2 ways another could have been second, and 1 way for the last one. So for any three selections, there are:
ways they could have been applied as to order. That means that of the 60 ways to select the toppings, there are only ways to select different toppings if the order they go on the cone doesn't matter.

: Natasha is allowed to duplicate her topping selections AND the order of application matters.

Again, when she selects the first topping, there are 5 choices. But when she goes to select the second topping, she again has 5 choices. And the same thing for selecting the third topping. Therefore:

ways to select the toppings.

: Natasha is allowed to duplicate her topping selections BUT the order of application doesn't matter.

This one is the most complicated calculation. Let's break it down into situations. First, we know that there are 125 ways to choose allowing duplication where order matters. Of those 125 ways, 60 of them are accounted for where all three toppings are different (this was our first way calculation), so that means that the remaining 65 ways must have some duplication. Of those 65 ways, 5 of them are the 5 different ways she could select three applications of the same topping. The remaining 60 ways are the situations where she selected two of the same topping and one different one.

Let's investigate the number of different orders that 2 of the same plus 1 different could be applied. Since there are three toppings, the 1 different one could be applied either first, second, or third. So that means there are three different order of application possibilities for any 2 + 1 selection available in the 60 ways to select 2 + 1. Therefore, for these 60, if order doesn't matter there are possible selections.

We know that for the 60 selections where the toppings are different, there are only 10 when order doesn't matter, and we also have the 5 possibilities for making all three toppings the same.

So now all we have to do is add these three sets of possibilities: 20 plus 10 plus 5 = 35 ways to select if duplication is allowed and order doesn't matter.

Hope this helps,
John

ATTENTION: Opportunity for Super-Double-Plus Extra Credit! How many ways would I have had to do this problem if there was a third un-answered yes/no question relating to the conditions of the problem?

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