Question 116173: A restaurant offered salads with 4 types of dressings and 7 different toppings. How many different types of salads could be offered?
Found 2 solutions by checkley71, solver91311:
Answer by checkley71(8403)   (Show Source):
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website! That depends. Can you have more than one type of dressing and/or more than one type of topping? If not, the answer is trivial: There are four ways to pick the dressing and seven ways to pick the topping, 7 X 4 = 28 -- assuming that you must pick 1 dressing and 1 topping. If you were allowed to either not have any dressing or topping or to just have a plain salad, then you would have 5 dressing choices (one of them being no dressing at all) and 8 topping choices, so you would have 5 X 8 = 40 possibilities.
Other situations are more complex to calculate. For example, if you are allowed two dressings and three toppings, you would first have to decide whether you could also have less than two dressings and three toppings, and whether the order in which the dressings and toppings went on the salad makes a difference or not.
So let's set some rules and do a calculation so you can see how it is done and perhaps extend the method to more general cases.
Let's say the rules are:
4 types of dressing
7 types of topping
You may have either 0, 1, or 2 dressings, but you may not duplicate (i.e. if you pick two dressings, they have to be different)
You may have either 0, 1, 2, or 3 toppings, but you may not duplicate
The dressing goes on first, but the order in which multiple dressings or toppings go on the salad makes no difference.
There is only one way to select no dressing at all, plus
There are 4 ways to select just one dressing, plus
if you select 2 dressings, there are 4 ways to pick the first one times 3 ways to pick the second one divided by 2 because for every possible pair of dressings, there are two ways for one to be first and the other second, and we said order of application doesn't matter:  ways to pick your dressing.
There is only one way to select no topping at all, plus
There are 7 ways to select just one topping, plus
if you select 2 toppings, there are 7 ways to pick the first one times 6 ways to pick the second one divided by 2 because order doesn't matter, plus
if you select 3 toppings, there are 7 ways to pick the first one times 6 ways to pick the second one times 5 ways to pick the third one divided by  because order doesn't matter, so  ways to pick the topping.
And finally,
 different types of salad.
You see why you have to be specific when you ask a question?
Hope this helps,
John
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