How many possible combinations/permutations?

Question

I have 104 ingredients, and there are a maximum of 3 ingredients, how many recipes can be made? Take into account the order matters, that makes this a question of possible permutations instead of combinations right? ex. apple + cherry + strawberry and strawberry + apple + cherry would be two different recipes. Also, flavors can be repeated, so cherry, cherry+cherry, and cherry+cherry+cherry would all be different recipes.

Answer 1

Hoe many 3-letter 'words' are possible with an alphabet having $n$ letters? aaa, aba, abc, acb etc are all to be treated as separate words. It is simply $n^3$, as first letter can be any of the $n$, second can also be any of the $n$ letters and so on. So $104^3=1124864$ is the number of recipes possible.

Can someone learn to make quarter as many of these dishes? Assuming one can learn each of them in half an hour there are enough recipes to learn continuously for 64 years (without allotting time for sleeping, playing, or shopping for those ingredients or for eating)

Answer 2

The recipe can have $1, 2\; or\; 3$ ingredients.

With repeats allowed, and order important,

number of recipes = $104 + 104^2 + 104^3$