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The total number of subsets of a set of 6 elements is.

1. 720
2. $6^{6}$
3. 21
4. None of the above

Total subsets are $\,^6C_0 + \,^6C_1 + \,^6C_2 + \dots + \,^6C_6 = 2^6 = 64$

Another way to look at it is:

To make a subset from the original set, we have 2 choices for each element. Each element can either be in the subset, or not be in the subset.

Therefore, total possible ways $= \underbrace{2 \times 2 \times \ldots \times 2}_{6 \text{ times}} = 2^6 = 64$

option D