7 Drinks - 7 Flavors - Infinite variety?

Solution 1:

This may be one of those times where it's important to distinguish between the mathematical and the physical.

If you are allowed to choose arbitrary ratios, then mathematically, the number of flavors are infinite. This is true even with two flavors, $a$ and $b$. If we let $0$ denote the flavor $a$ and $1$ denote the flavor $b$, then the line joining $a$ and $b$ is just the unit interval, which has uncountably many points. If each point represents a unique flavor, it follows that there are an infinite number of flavors.

In reality though, the number of flavors is finite for two important reasons. First, the human tongue cannot distinguish flavors to arbitrary accuracies. If nobody on Earth can tell the flavors $0.499$ and $0.5$ apart, do you still call them different flavors? Secondly, there are only a finite number of atoms which fit into the cup so the flavors are trivially bounded by the combinations of atoms you can choose to make the drink, which is finite.

Solution 2:

The answer is no, because of limitations in the physical world. In a container of given size, a limited number of soda molecules will fit.

For example, assume you have a very small cup only capable of containing $100$ molecules of soda. (To simplify the problem, assume all soda molecules are the same size.)

So, you have $100$ "slots" for soda molecules. Each slot can have one of up to $7\cdot7=49$ possible flavors. Thus, you only have $49^{100}$ possible combinations of flavors--a large number, for sure, but still a long ways off from infinite.

EDIT:

The limited capacity of the cup is important. If we had no limit of capacity, we could form any ratio we so wished between any mixture of drinks. However, this is not possible with limited capacity. For example, let's say we wanted a ratio of $100:101$ molecules of soda (in the previous example). We'd need a total capacity of $201$ molecules to have this precision. Unfortunately, we only can fit $100$ molecules in a cup.

Solution 3:

Yes, if you can control the ratios to an arbitrarily (even if finite) precision then you can make infinitely many different drinks with only two flavours available. Simply take $p$ such that $0<p<1$ and take $q=1-p$. Now take $p\times 500\text{ml}$ of the first flavour and $q\times 500\text{ml}$ of the second flavour. There are infinitely many $p$'s between $0$ and $1$ so there are infinitely many combinations.

The surprising thing, if anything, is that with seven flavours or with two flavours -- the number of different drinks you can generate is the same. And the same means not just "infinite" but rather a very good mathematical notion of "same size" for infinities.

Do note, however, that this is all theoretical in an ideal universe of mathematics. Our [physical] mouths don't have arbitrary precision between flavours and we can't control the machine more than a finite and limited precision, moreover there is only a finite amount of drinks we can make during the lifetime of ours and of the machine. So there is only a finite number of combinations in reality from which we can choose after all.