# Making sense of combinatorics-based marketing hyperboles

Diablo 3 has 97 billion possible skill/trait builds. Per class.

LessPop_MoreFizz, emphasis is mine.

I used base two logarithm to claim "97 billion" configurations only are roughly 37 binary choices because $\log_2(97\times10^9) \simeq 36.5$, but I would like to know if there are more refined techniques to remove the marketing out of these big numbers and see their true meaning.

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Dear LessPop, I didn't mean to put your name there to accusingly point a finger in your general direction -- it's merely required by the attribution requirements of the chat content license. – badp Oct 26 '10 at 10:38
And the Rubik's cube has more than "3 billion combinations" :-) – Robin Chapman Oct 26 '10 at 11:03
(For those who don't know, Robin is referred to a frequently quoted but totally incorrect number. The actual number of positions is about 4.3 x 10^{19}.) – Qiaochu Yuan Oct 26 '10 at 13:02
(... and roughly 12 times more @ 5.19e20 if you allow complete disassembly of the toy.) – Willie Wong Oct 26 '10 at 14:10

It's right and wrong.

There is a certain raw number of possibilities and let's assume from the earlier remarks that 97 billion is the correct number.

What the hype does not mention is that there is also some statistical distribution to these possibilities. Some are much likelier than others, and some selections with different "genotypes" have similar "phenotypes": they are almost the same in effect. The amount of information in one selection from the 97 billion is, on average, a lot less than the number of bits needed to specify a unique integer between 1 and 97000000000.

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I think this is a fine approach. Some of the choices could be three-way, but to get a general idea of how many choices it takes to get that many total possibilities you are spot on.

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