Capsaicin Overload!

If we make chili sauces that use every possible combination of any number of the chili types we have, we are making $X$ binary choices when we design a specific sauce. Say that there are $X$ kinds of chili and order them $1-X$ . Then, for each kind of chili, choose to either Include It = 1, or Do Not Include It = 0.

For example 10000...000 would represent the sauce that contains the first chili type and no others, and 11111...111 would be the sauce that contains every chili type.

The number of chili sauces will therefore be $2^X$ by the product rule. If $2^X = 8192$ , then $X=\fbox{13}$ .

So the chili sauces you own are a set. The set contains all the chili sauces with none, 1, 2, 3, etc.chilis in them. In other words, this is a power set. If you have a set, and want to know how many subsets there are, then for each element in the set you have a binary choice: either you put it in the subset you are creating, or not. And so, the formula for calculating the number of subsets is 2^n, where n is the number of elements in your set. Here n, the number of chilis, is an unknown. But, you have been given the answer though: 2^n = 8192.

In short, n = log 8192 / log 2 = 13

Well, for every chili it can be in the sauce(1) or not in the sauce(0), so 2 possibilities. With 13 chilies it would then be 2^13 = 8192 posibilities. In order to be a chili sauce it has to contain at least 1 kind of chili. But 1 of the combinations has 0 of all the chilies and is therefore not a chili sauce. So the correct number of distinct chili sauces is 8191.

If she has 8192 bottles with chili sauce, and there is only 8191 different chili sauces, at least 2 bottles contains the same combinations of chilies (pigeon hole theory), so the can't all be unique combinations.

Let N be set containing all the chillies tgen 8192=2^X or X=13 as it is the no. Of subsets of N!!!