Combinatorics
:back to Mathematics -- Finite Mathematics
It may seem surprising that the number of possible orderings of a deck of 52 playing cards is 8.065817517094e+67. That is a little bit more than 8 followed by 67 zeros. Comparing that number to some other large numbers, it is greater than the square of Avogadro's number, 6.022e+23, "the number of atoms, molecules, etc. in a gram mole".
That large number, 52 factorial, is the product of all the natural numbers from one to fifty-two, the number of different orders the deck can have after shuffling. Calculating the number of ways that certain patterns can be formed is the beginning of combinatorics. Some very subtle patterns can be developed and some surprising theorems proved.
One example of a surprising theorem is that of Frank P. Ramsey which essentially says (in mathematical language) that if you look hard enough, any pattern of stars can be found in the sky. It has been used to debunk claims that some patterns are especially meaningful.
We assume the existence of a Set S of N objects. Combinations of r objects from this set S refer to subsets of S (where the order of listing the elements does not distinguish two subsets). Permutations of r objects from this set S refer to sequences of elements of S (where two sequences are considered different if they contain the same elements but in a different order).