An iPhone 7 takes pictures that have roughly 12 Megapixels. For simplicity, let's assert that the picture only encodes 256 values per red, green and blue channels such that a 1x1 pixel image has 256^3 unique representations (permutations). The number of permutations for the entire 12 Megapixel image is 256^(3 * number of pixels) = 256^(3 * 12,000,000) = 256^(36,000,000).
There are about 3 billion nucleotide bases in the human genome (which can be represented as a long string that looks like "ACATGACTTGAT..."). Since there are 4 bases by which all our DNA is expressed, the number of permutations in the human genome is 4^3,000,000,000.
For comparison, the estimate of atoms in our observable universe is 10^80
What's bigger: 4^(3,000,000,000) or 256^(36,000,000)?
I tried a couple BIG NUMBER calculators online (like this one from CASIO) but none of them seem to be able to handle numbers this big.
So if I try comparing their logarithms I get:
4^(3,000,000,000)
= 4 ^ (3 * 10 ^ 9)
= (3 * 10 ^ 9) * log(4)
= (3 * 9 * log(10)) * log(4)
= 27 * log(10) * log(4)
≃ 16
256^(36,000,000)
= 256 ^ (36 * 10 ^ 6)
= (36 * 10 ^ 6) * log(256)
= (36 * 6 * log(10)) * log(256)
= 216 * log(10) * log(256)
≃ 520
Did I do that right? If so, can it be said that there are about 33x more permutations in an iPhone picture than there are in the human genome?
Update
My math was indeed quite poor. Here's an update
4^(3,000,000,000)
= 4 ^ (3 * 10 ^ 9)
log (4 ^ (3 * 10 ^ 9))
= (3 * 10 ^ 9) * log(4)
≃ 1806179974
256^(36,000,000)
= 256 ^ (36 * 10 ^ 6)
log(256 ^ (36 * 10 ^ 6))
= (36 * 10 ^ 6) * log(256)
≃ 86696639
So the number of permutations in the data that describes the human genome sequence, though most of them will not yield a human, is much larger than the permutations of pixels in an iPhone image (regardless of how imperceptibly different those permuted images may be). I was tempted to try to find the ratios between these two logarithms again but realized that wouldn't be representative of the actual quotient. The actual quotient is:
(2^(6,000,000,000))/(2^(288,000,000))
= 2^(6,000,000,000 - 288,000,000)
= 2^(5,712,000,000)
So the data in the human genome has 2^(5,712,000,000) more permutations than the data in an iPhone picture.