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

Why do we have the "minus 1"?

Just answer the question: What is the maximum value of an 1-bit integer?

One bit integer can store only two (2¹) values: 0 and 1. Last value is 1₂ = 1₁₀

Two bit integer can store only four (2²) values: 00, 01, 10 and 11. Last value is 11₂ = 3₁₀

Thus, when integer can stores N, values last value will be N-1 because counting starts from zero.

n bit integer can store 2n values. Where last will be 2ⁿ-1

Example: One byte can store 2⁸ (256) values. Where first is 0 and last is 255

Why isn't the maximum just 2n?

Because counting starts from zero. Look at first value for any n bit integer.
For example byte: 00000000

This would be very confusing if:
00000001 means 2
00000000 means 1

would not? ;-)