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solipsys.co.uk

A short time ago someone posted[0] about just how big 52! is, so I thought I’d share this little nugget

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Here’s the trick I would use if I needed to calculate a factorial and didn’t have a computer on hand. It has a powers of 2 bias, so lg x refers to the base-2 logarithm.

7! = 5040 ~ 2^12

lg 8 + lg 9 + ... lg 15 has 8 terms of “about 3.5”, so is about 2^28. lg 16 + ... lg 31 has 16 terms of “about 4.5”, so it’s about 2^72. lg 32 + ... lg 52 has 21 terms of “about 5.33”, so it’s about 2^112.

So you get a total of around 2^224. The actual answer is 2^225.56..., so it’s not bad for an approximation can be done on a bus without a calculator.

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Actual error bound?

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Someone else has taken the title literally and computed the error when you divide by 3000 instead of (54x53). However, that’s not the point of the article.

You can get a handle on the error in the article by identifying all the approximations and estimating from there. I’ll be doing that in a follow-up, but the main error comes from using e ~ 2.7. It’s closer to 2.72. That error is about 0.7%. When we raise to the power of 54 that gives an error of about 40% to 50%.

But that is offset by the approximation that 2^10 ~ 10^3, and raising that to the power of 5. It’s an interesting and useful exercise to refine the original by chasing down estimate of the errors. I might do that in a follow-up post.

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By the definition of factorial as 1×2×…×n, you can get 52! exactly if you divide 54! by 53×54=2862. If you divide by the approximation 3000 instead of 2862, the result is exactly 95.4% of 52!.

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There’s a great deal more approximation being done than just 3000 ~ 2862. The calculation of 54! is itself very approximate.

Anyway, now that I’m back at my laptop: (10^68 - 52!) / 52! ~ 0.24.

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I really hope you understand that that’s not the point of the article.