A has:

1 subset with zero elements = ∅

4 subsets with one element = {a}, {b}, {c} and {d}

6 subsets with two elements = {a, b}, {a, c}, {a, d}, {b, c}, {b, d} and {c, d}

4 subsets with three elements = {a, b, c}, {a, b, d}, {a, c, d} and {b, c, d}

1 subset with four elements = {a, b, c, d}

Therefore A has 16 subsets altogether.

Note:

1. Remember that order doesn't matter. So, for example, {a, b} and {b, a} are the same subset.

2. The numbers 1, 4, 6, 4 and 1 are Binomial coefficients and occur in the fifth row of Pascal's triangle.

3. 16 = 2⁴. Generally a set with n elements has 2ⁿ subsets.

I found an answer from www.quora.com

If A= {a, b,c, d}, then how many subsets does the set A have? - Quora

The number of subsets of a set is 2^n, where n is the cardinality of the set. D has 4 elements. So it has 2^4 = 16 subsets, including The Empty Set and D itself.

For more information, see If A= {a, b,c, d}, then how many subsets does the set A have? - Quora

I found an answer from en.wikipedia.org

Power set - Wikipedia

In mathematics, the power set (or powerset) of a set S is the set of all subsets of S , including the ... For example, the power set of a set with three elements, has: ... the singleton subsets),; C(3, 3) = 1 subset with 3 elements (the original set itself). ... However, there are two important properties of subsets that do not carry over to  ...

For more information, see Power set - Wikipedia