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I've come across the following simple, but unexpected equality numerous times accidentally.

$$ \frac{10!}{6!} = 7! $$

which is the same as

$$1*2*3*4*5*6*7 = 7*8*9*10$$

Does it hold any specific (geometric?) meaning? I've graphed out numerous similar identities, but didn't find any that have an unexpected result similar to the above (leaving aside the trivial $\frac{n!}{(n-1)!} = n$).

Zev Chonoles
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Etheryte
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1 Answers1

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If you're looking for some explanation as to why the two factorial expressions are equal, just break each expression down into prime factors. $$1*2*3*4*5*6*7=7*8*9*10$$

$$1*2*3*(2^2)*5*(2*3)*7=7*(2^3)*(3^2)*(2*5)$$

$$1*2^4*3^2*5*7=1*2^4*3^2*5*7$$

I know this only scratches at the surface of your questions about it, but I hope you still find it of some value.

Leonidas Lanier
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