How many total bits are required to represent 512 different numbers?

To determine how many bits are required to represent 512 different numbers, we can use the formula for calculating the number of distinct values that can be represented with a certain number of bits. Specifically, if ( n ) is the number of bits, then the total number of unique values that can be represented is ( 2^n ).

In this case, we need to find the smallest ( n ) such that ( 2^n ) is greater than or equal to 512.

Calculating powers of 2, we find:

• ( 2^8 = 256 ) (which is not sufficient)

• ( 2^9 = 512 ) (this works)

Since ( 2^9 ) equals 512, it indicates that 9 bits are exactly needed to represent all different combinations for 512 distinct values. Therefore, the correct answer is that 9 bits are required to represent 512 different numbers.