An even stronger result takes into account the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem http://en.wikipedia.org/wiki/Birthday_problem).

An even stronger result takes into account the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem http://en.wikipedia.org/wiki/Birthday_problem). Claim: No two properly shuffled decks have ever been the same. Proof: For convenience, set upper bound on # of shuffles = n = 10^23 and number of possible shuffles = N = 10^68. Then the probability that all shuffles are not unique is q = 1 X (1 - 1/N) X (1 - 2/N) X ... X (1 - (n-1)/N). Since n << N, even (n-1)/N is small, so we can approximate q as q = 1 X e^(-1/N) X e^(-2/N) X ... X e^(-(n-1)/N) = e^(-n^2/(2N)) using the series expansion e^x = 1 + x for x << 1. Then the probability that any two decks have ever matched is p = 1 - q = 1 - e^(-n^2/(2N)). Now, q = e^(-5 * 10^(-22)) = 1 - 5 * 10^(-22), to good approximation, so p = 1 - q = 5 * 10^(-22) which is zero for all practical purposes. Thus, no two properly shuffled decks have ever been the same. QED