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Somone has suggested that

Within, say, a collection of every possible 30 second long MP3 file encoded at 128kbps, I'd probably be infringing on a few thousand copyrighted works.

128kilobits per second = 128,000 bits per second * 30 seconds = 3,840,000 bits.

There are 2 to the 2,840,000 possible files of that length.

Ignoring the fact that most of those won't be valid mp3s, how can I quantify the amount of space needed to store all those files? For instance, is that more bits than there are atoms in the universe?

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    The first equation has the units incorrect. It is not 128,000 bits, it is 128,000 bits/second. The error is corrected across the second equal sign. Then a typo in the next line.2012-10-30

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The total space is $2^{3,840,000}\cdot 3,840,000$ bits. This is a fine quantification, but if you want the number of zeros in base $10$, you can take the log of it. $\log_{10}(2^{3,840,000}\cdot 3,840,000)=3,840,000 \log_{10} 2 + \log_{10} 3,840,000 \\ \approx 3,840,000\cdot 0.30103 + 6.58433 \approx 1,155,961$

so you need a little more than a million zeros.