Is it possible to have infinite combinations in reality? On a yoghurt advert, the voiceover claimed that you have infinite combinations with it. However, given that there is a finite amount of matter, is it possible to have infinite combinations with the yoghurt or just a very large number of combinations?
 A: All possible combinations or permutations of a finite amount of "stuff", will always be finite. This is a basic fact from combinatorics.  
Details
Number of ways to select $k$ things from a set of $n$ things:
$\mathcal{C}(n,k)=\frac{n!}{k!(n-k)!}$, where $m!$ means $1\times2\times3\dots m$. Reference. Here order does not matter
Number of ways to permute $k$ things from a set of $n$ things: $\mathcal{P}(n,k)= \frac{n!}{(n-k)!}$. Reference. Here order does matter
These expressions hold for $k\le n$.
A: I accept Mustapha's point for discrete systems, the problem is that  we don't know if the amount of matter the universe is discrete (i.e. finite combinations) or continuous (i.e. infinite combinations).
Say you added differing ranging amounts of food items. For example, apple, strawberry, and kiwifruit. Then, to me at least, there is no limit to the different potential flavors, but I doubt if you could tell the difference in taste caused by minute differences in concentration of the ingredients. 
As far as finite amount of matter goes, who knows how much matter the universe contains, we can't test it, so we don't know.
And the ultimate discrete amount, to push it to the limit, is the number of atoms in the ingredients, but we don't know how many apples, strawberrys etc the universe actually contains.
 
A: Whether you have a finite number of pieces is not the important part, it is the relationships they may have to one another.
Consider a point proton and an electron with a perfect $1/r$ potential without second quantization and at zero Kelvin (no temperature). There are an infinite number of energy eigenstates, so there are an infinite number of different relationships between the two objects.
So any argument that merely counts the number of things without counting how many ways they relate to each other is essentially incomplete.
But I had to assume many things above to get those infinite number of distinct states.  If you assume that temperature is never perfectly zero, then you can imagine a (possibly small, but finite) spread of energies that can't really be kept separate.  Also you can't just go to arbitrarily high energies (because of black hole densities but also from pair creation playing havoc with the idea of a finite amount of stuff).  And from stability you want an energy minimum.  So take the energy minimum subtract it from the energy maximum and divide by the smallest width of the energy spread and that's really a finite band of energies.
So you can try to partition the energy amongst the different parts an it looks like stability, keeping a finite amount of matter, and thermal lack of control constrain your choices to a finite number of controllable (thermal) allowed (not too energetic) accessible (not breaking stability, so for instance we could consider iron as the stable nucleus) interactions.
A: After adding a large but finite number of atoms, the yoghurt will collapse into a black hole.  After that point, the flavour will not change, no matter what new atoms you add, because Black Holes Have No Hair.
So, no, the number of combinations is finite.  On the other hand, it is large enough that there isn't enough space in the visible universe to write it down in. :-)
A: first of all your question seems to me very ambigous in technical context of physics and mathematics after all what type of combinations you are talking about regarding to yogurt?if we want to try combinations of yogurt with many things like :
yogurt with salad,frozen yogurt ice cream,yogurt beverages like ayran ,greek yogurt based dish tzatziki( i think these were the things ad was saying about ) 
As far as combinations are concerned of one object with other the formula stated  by Mustapha Mond is correct or simply there is no mathematical basis for combination of finite(by all dimensions like size mass length everything) object with infinite object 
so not possible atleast mathematically not possible! 
