Interview Cake
Let's start simpler: how many integers can we express with 1 bit? Just 2:
How many can we express with 2 bits? Well for each possibility for the first bit, we can put a 0 after it or we can put a 1 after it:
So by adding a second bit, we have twice as many possibilities as with just 1 bit. 2 * 2 = 4 possibilities in total.
Same idea with 3 bits—for each of the 4 possibilities with 2 bits, we can put a 0 after or we can put a 1 after. That gives us twice as many possibilities as with 2 bits, which is twice as many as with 1 bit, so 2 * 2 * 2 = 8:
Do you see the pattern? In general, with n bits, we can express 2^n different possible numbers!
So with 8 bits, that's 2^8=256 different numbers.
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