How many 32 digit binary number combinations are possible?
For example:
$$00000000-00000000-00000000-00000000$$
$$00000000-00000000-00000000-00000001$$
$$00000000-00000000-00000000-00000010$$
$$.$$
$$.$$
$$.$$
$$11111111-11111111-11111111-11111110$$
$$11111111-11111111-11111111-11111111$$
Well, we can easily convert the last number to decimal to get the number of combinations and the answer would be $2^{32}$ combinations. But I need a general explanation with respect to general concepts like probability, permutations or combinations, as to how we get $2^{32}$ combinations.
P.S. I am not a mathematician. So, please try to explain in a simple way. Thanks a lot!
Best Answer
For each bit (binary digit) that you have, there are two possibilities: Either it can be a zero, or it can be a one.
Therefore, if you have one bit, you have two possible numbers. If you have two bits, each of them can be either a zero or a one, and since there are two possibilities for the first, and two possibilities for the second, there are $2^2 = 4$ total possibilities.
Similarly, if you have some number $n$ of bits, each of them can be a zero or a one, and there will therefore be $2^n$ possibilities.