Max Mantissa $2^{bits}-1$

Question

if we look at a $5$ bit mantissa, the max value will be $11111$ which is $2^5-1$, Why is it in the form of $2^{bits}-1$ is it a combinatorial explanation?

Answer 1

If you look at unsigned binary integers, the smallest in $n$ bits is $00000$ with $n$ zeros and the highest is $11111$ with $n$ ones. It is just like in base $10$ where the largest number with $n$ digits is all $9$s. Then to see that this equals $2^n-1,$ note that the bits are successively $2^0+2^1+2^2+\ldots +2^{n-1}$. The sum of this geometric series is $2^n-1$