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The power set will have $2^c$ elements.
This is because while generating the powet set, we have two choices for each element in the original set.
Hence, $\underbrace{2 \times 2 \times \dots 2}_{\text{c times }} = 2^c$.
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One well know way is already there in answer. Here is another way.

The number of subsets with k elements in the power set of a set with n elements is given by the number of combinations, C(n, k), also called binomial coefficients.

For example, the power set of a set with three elements, has:

  • C(3, 0) = 1 subset with 0 elements (the empty subset),
  • C(3, 1) = 3 subsets with 1 element (the singleton subsets),
  • C(3, 2) = 3 subsets with 2 elements (the complements of the singleton subsets),
  • C(3, 3) = 1 subset with 3 elements (the original set itself).

Using this relationship we can compute  using the formula:

 

 

Therefore, one can deduce the following identity, assuming ${\textstyle |S|=n}$​​​​​​​

 

 

 

Source: https://en.wikipedia.org/wiki/Power_set#Relation_to_binomial_theorem

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