Total Number of Possible Combinations

[tostenson]

Hello, I am a graphic designer working on a menu for a friend of mine and math was never my strong suit. Also, add the fact that I've been out of school for some time and don't do math regularly. Anyway, I have a pretty simple question that I could use some help with if someone doesn't mind.

I'm wanting to add a "total number of possible combinations" to his menu, since he has quite a few options.

So here's my query:

He has 10 types of meat
He has 65 different types of toppings

How many different combinations can he make out of all these items?

Thanks in advance for the help :)

 

[Mark44]

Can you narrow the problem down some? Would it be possible to order a pizza (I assume that's what you're talking about) with all 10 types of meat and all 65 types of toppings? Or are there limits on the numbers of types of meat and toppings?

 

[LCKurtz]

If you are wanting to brag about the large number of possible topping choices available, you answer is 2⁷⁵ = 37778931862957161709568. This allows the choices of "everything" or nothing and everything in between.

 

[tostenson]

Sure to help narrow it down, he has a hotdog restaurant, which has 10 types of sausage choices and 65 different toppings. Though, I think LCKurtz nailed my question. The point was to say "you can choose from ___ different flavor combinations". Thanks :)

 

[Mark44]

Sure to help narrow it down, he has a hotdog restaurant, which has 10 types of sausage choices and 65 different toppings. Though, I think LCKurtz nailed my question. The point was to say "you can choose from ___ different flavor combinations". Thanks :)

But I'm guessing that you can't order a hot dog with, say, four hot dogs on it, so you won't get all of the combinations that LCKurtz listed.

 

[tostenson]

You're right Mark. So the formula would be 2 to the 66th power then?

 

[Dr. Seafood]

You're right Mark. So the formula would be 2 to the 66th power then?

I think so.

 

[Mark44]

You're right Mark. So the formula would be 2 to the 66th power then?

I think so.

I don't think so. Let's work with the toppings first. Since there are 65 of them, and anyone of them could be included, there are 2⁶⁵ different combinations of the toppings. In case that's not clear, let's suppose there is just one topping, and it is either put on or not. That's two combinations for the one topping, which happens to be 2¹.

Now let's suppose there are two toppings, A and B. The different combinations are:

(no topping)
A
B
A and B

That makes four combinations for the two toppings, which happens to be 2².

In general, if there are n toppings, there will be 2ⁿ combinations of the toppings. For the 65 toppings, this means there are 2⁶⁵ possible combinations. This runs the gamut from no toppings at all, all the way to all 65 of them being on the hot dog, as well as every intermediate possibility.

OK, now let's figure out the combinations with a sausage included. Since there are 10 different sausages, and we are assuming that a hot dog won't have more than one sausage, for each choice of sausage there are 2⁶⁵ ways we can top it. That makes 10 * 2⁶⁵ different hot dog combinations.

If we also include the possibility that some doesn't want the sausage included, that makes 11 * 2⁶⁵ different combinations, one of which would be a hot dog with no sausage, and no toppings. IOW, just a bun.