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Here are some charts that shows all permutations of the binary number system limited to one, two and three digits in length. Thus the number of possible combinations are $2^1, 2^2$ and $2^3$ respectively. This demonstrates that the number of permutations for a binary number of $n$ length is $2^n$. It is easy to see here that the number of possible combinations of $1$s and $0$s is much greater then the number of digits in the number; 1 digit numbers have 2 possible combinations, 2 digit numbers have 4 possible combinations, 3 digit numbers have 8 possible combinations, etc., etc., etc. In fact, as the number of digits $n$ grows bigger the possible combinations grow exponentially by $n$. Eventually, as $n$ approaches infinity the difference between the number of digits in a binary number and the number of combinations those digits can represent is far, far greater then $n$. It should be $2^nn$. For example: $21=1, 42=2, 83=5$. Even the difference grows faster then $n$. However, when $n$ actually is infinity, that is, a binary number is infinite in length, all bets are off and they all cross the finish line neckandneck. Everything is infinite. The size of the number versus the number of numbers becomes square. For a chart that looks something like:
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This chart has been drawn in a square configuration, the number of digits is represented horizontally and the number of combinations is on the vertical, to illustrate the above fact that there are the same number of digits to the left and right as there are up and down, both reaching to Infinity.
Let it be noted here that Infinity is not a number. There is no value attached to any quantity of infinity. Infinity is more a class denoting the uncountability of a value. As such, any portion of that value is equally uncountable unless the portion that is taken from it has been specified. For example: An infinite value  1. Now, of course, the 1 is countable but the remainder is still just as uncountable as the starting value. In fact the starting value has not been affected at all.
What about for another number system? Say, a decimal system.

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Here the number of permutations versus the length of the number is $10^n$ for a number that is $n$ digits long. Compared to the binary system; for a number of $n$ length, the $10^n$ values for a decimal system are a lot more then the $2^n$ values for a binary number of the same length. In this case as $n$ approaches Infinity, the number of permutations or values representable is astronomically more then for a binary digit of the same length. I mean, like dude, take $10^{100}$ versus $2^{100$ and compare the difference then try $10^{500}$ versus $2^{500}$ and $10^{1000}$ versus $2^{1000}$ and compare their differences. What is the trend here? Well, $10^{100}$ is represented on my calculator as $10^{100}$ (We know, however, this number is 101 digits long) but $2^{100}$ is $1,267,650,600,228,229,401,496,703,205,376$ (31 digits long) which in terms of scientific notation equals a bit less then $1.3 x 10^{30}$. That’s a large difference, right? But these things have to be read in context or by comparison to something else. Now try $10^{500}$ versus $2^{500}$. Now we have a number we can sink our teeth into. $2^{500} = 3.3 x 10^{150}$. That is a big number but the decimal number is still much bigger. In fact, the difference has grown even greater. What about the last set of numbers I mentioned earlier? $10^{1000}$ versus $2^{1000}$. $2^{1000}$ equals $1.1 x 10^{301}$. That has the biggest difference yet. We can notice that the proportion of the exponents remain much the same but the differences between the two numbers is growing by that proportion. Will the value of a decimal number therefore reach Infinity before the value of a binary number as n goes to Infinity? Will the number of permutations so far out strip the number of digits between the decimal number and the binary number that the decimal number’s value reaches Infinity before the binary number’s digits n reaches Infinity? It is clearly approaching Infinity faster, isn’t it? The fact remains that all values that head towards the Magical Realm Infinity will reach it at the same time, even though order is maintained. The Cops on the road to Infinity are very insistent about maintaining order. However, once the Realm of Infinity has been reached Chaos is the rule. It’s kind of like how once you square a negative number you loose the negative property and no amount of square rooting can get it back. Oh yeah, I remember now. We made up a number that allows us to do that. We just imagined there was such a number and POOF! i. WHOA! 
I don't know about you but doesn't there seem to be something strange about a concept that at once has no value and a universal value of equality for all members?