What is the result of using 64-bit integers instead of 32-bit integers for counting purposes?

Using 64-bit integers instead of 32-bit integers significantly increases the range of values that can be represented. A 32-bit integer can represent (2^{32}) distinct values because with 32 bits, each bit can be in one of two states (0 or 1). This means that the maximum number of unique combinations you can create with 32 bits is (2^{32}), which is 4,294,967,296 possible values.

When you switch to 64-bit integers, the number of distinct values that can be represented increases to (2^{64}). This represents a much larger range, specifically (2^{32}) times more values than a 32-bit integer. Therefore, the correct choice reflects that the transition from 32-bit integers to 64-bit integers allows for (2^{32}) times as many values to be counted.

This significant increase in the amount of data that can be represented makes 64-bit integers particularly advantageous in computing for counting large items, managing complex datasets, and ensuring that a program can handle larger values without encountering overflow errors.