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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?

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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 $k\le n$.

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  • $\begingroup$ This answer doesn't take into account the fact that particles can be created and destroyed. The total number of particles is not fixed (see quantum field theory.) $\endgroup$ – Kenshin Jun 20 '15 at 23:50
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    $\begingroup$ We don't need to know the precise number of particles. We can obtain an upper-bound, which will still be finite. The upper bound can be obtained if you assume that the created number of particles is still finite. $\endgroup$ – Mustapha Mond Jun 21 '15 at 0:10
  • $\begingroup$ If I'm not mistaken, Space-Time is assumed continuous, so calculus can work, for the majority of Field theories. So, the possible orientations or combinations is infinite because the underlying Space has an infinite level of detail. $\endgroup$ – Zach466920 Jun 21 '15 at 16:37
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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.

 

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  • $\begingroup$ Is it physically possible for there to be infinite matter in the universe though? $\endgroup$ – Beta Decay Jun 20 '15 at 20:54
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    $\begingroup$ I would say we just don't know, we can only guess. If anybody told me the universe was finite, I would ask, how do you know? $\endgroup$ – user81619 Jun 20 '15 at 20:56
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    $\begingroup$ Interesting fact: one can calculate that there are approximately $10^{80}$ particles in the observable universe. Reference. That said, @AcidJazz is correcting is stating that we do not know whether the universe is finite or infinite -- either one of the outcomes lead to come really wacky conclusions about the nature of reality! $\endgroup$ – Mustapha Mond Jun 20 '15 at 23:26
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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.

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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. :-)

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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!

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    $\begingroup$ This is more suitable as a comment rather than an answer $\endgroup$ – Beta Decay Jun 21 '15 at 10:52

protected by Qmechanic Jun 21 '15 at 11:38

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