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If I have 16 different symbols, how many of these symbols can I combine to create no more than 16.777.216 unique combinations?

Eg. I have all symbols of the Hexadecimal language (16).

How many combinations can I create with X amount of symbols to not have more than 16.777.216 unique combinations maximum?

The position of each symbol is of importance.

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How many combinations do you get with 2 symbols? How many with $n$ symbols for $n\in\mathbb{N}$? – Alex B. Dec 4 '10 at 15:56
up vote 4 down vote accepted

Six, because $16^6 = 16.777.216$.

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-1. Somehow, I doubt that the number was what the poster needs. Rather, he would be much better served if he was empowered to solve such questions himself, which can only be done by practice. – Alex B. Dec 4 '10 at 15:58
@Alex Bartel: I think that's presumptuous. If Tom just wanted a hint or a more general solution, he would have asked for it. He's not my student, and I feel it would be condescending of me to try to teach him something he's not asking to learn. – Alex Basson Dec 4 '10 at 17:13
@Alex Basson I think that the fact that he is somebody else's student is beside the point. How would you feel if your students went onto the internet and had the homework you set them done there by other people? You would get wrong feedback about their progress and they wouldn't learn anything. Moreover, the time you may have spent coming up with nice exercises for them (I can easily spend a day devising an exercise sheet) would be wasted. I think that teachers of mathematics should be pursuing a common goal and not undermine each other's work. I am sorry, if that seems presumtuous to you. – Alex B. Dec 5 '10 at 2:27
@Alex Bartel: I'm sorry I wasn't clear when I said "He's not my student." My point is that I have no reason to believe he's anyone's student, not just that he isn't mine. For all I know, he's a working professional who's just trying to get a job done and needs a quick answer. I'm not trying to undermine anyone's work, I'm trying to help Tom do his. It just seems presumptuous to me to assume he's a student at all; in the absence of other information, I chose to treat him like an adult just looking for some information. – Alex Basson Dec 6 '10 at 11:53
The statements about "what the poster needs", what "would much better serve him", what would "empower [him] to solve such questions himself", and the idea that this can "only be done by practice" are all presumptuous and highly debatable. The answer given is illuminating and may be exactly the hint needed to grasp the combinatorial idea involved. Practice is rarely useful when basic ideas are missing, it is an "empowerment" to run in circles. And it is not necessary to decide for the OP what is good for him or to impose such decisions on those who contribute on-topic answers. – T.. Dec 6 '10 at 12:32

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