# A deli offers 3 different kinds of bread and 8 different kinds of toppings on its submarine sandwiches. How many different sandwiches are possible?

If we're only putting on 1 topping per sandwich, 24. If we're allowed as many toppings as we want (assuming at least 1), 765.

#### Explanation:

This depends on whether the toppings are only put on singularly or potentially jointly.

For the bread, I'll assume that we're only going to use 1 kind of bread - so that's 3 choices of bread.

For the toppings, we could add 1 topping, and so that'd be 8 choices. With the 3 choices of bread, we have:

$3 \times 8 = 24$ different sandwiches

But only 1 topping on a sandwich is boring! I might want 5 toppings. Or all 8! Since we aren't told that only one topping is put on, we need to sum up the possible combinations, and so we'll end up with (I'm going to assume that we need to have at least 1 topping, otherwise we don't really have a sandwich but rather just a big hunk of plain bread):

$3 \times {\sum}_{k = 1}^{n} {C}_{n , k}$

$3 \times \left(8 + 28 + 56 + 70 + 56 + 28 + 8 + 1\right)$

$3 \times 255 = 765$ sandwich options