Why is the maximum value of an unsigned n-bit integer 2^n-1 and not 2^n?

Question

The maximum value of an n-bit integer is 2ⁿ-1. Why do we have the \"minus 1\"? Why isn\'t the maximum just 2ⁿ?

Answer 1

2^32 in binary:

1 00000000 00000000 00000000 00000000

2^32 - 1 in binary:

11111111 11111111 11111111 11111111

As you can see, 2^32 takes 33 bits, whereas 2^32 - 1 is the maximum value of a 32 bit integer.

The reason for the seemingly "off-by-one" error here is that the lowest bit represents a one, not a two. So the first bit is actually 2^0, the second bit is 2^1, etc...