How many arrangements of numbers is possible?

Answer 1

There are an infinite number of possibilities based on the infinite set of numbers.

However, for a finite set, there are limited possible combinations, depending on whether you can use the same numbers over again, or if they have to be distinct, or if their order makes any difference.

Here's an example:

For a group of THREE numbers, there is only one possible group of 3 numbers, and there are three possible groups of 2 numbers (i.e. 12, 13, 23) .

Using each of three different numbers, there are 6 ordered combinations of two numbers (12, 13, 23, 21, 31, 32) and 6 possible combinations of three numbers (123, 132, 213, 231, 312, 321).

If the numbers are allowed to repeat, there are 9 possible combinations of two (add 11, 22, 33) and 8 more possible combinations of three (111, 112, 113, 122, 133, 222, 223, 333) - if order matters, each triple (111) has only one possible order, each double has three (112, 121, 211).

The number rapidly increases for larger numbers of possible and larger groups from those sets. The possibilities are called combinations and permutations, and are connected to the numerical property called "factorials" (a number multiplied by all smaller integers - 2 factorial is represented by "2!" and equals 2 x 1 = 2, while 3! = 3 x 2 x 1 = 6). The number of discrete sets of K numbers from N possible numbers is N! / K! x (N-K)!