1. **State the problem:** We need to find how many subsets of the set $S$, where $S$ is the set of factors of 2026, contain exactly 2 elements. 2. **Find the factors of 2026:** First, factorize 2026 into its prime factors. $$2026 = 2 \times 1013$$ Since 1013 is a prime number, the factors of 2026 are: $$S = \{1, 2, 1013, 2026\}$$ 3. **Count the number of elements in $S$:** There are 4 elements. 4. **Number of 2-element subsets:** The number of subsets with exactly 2 elements from a set of $n$ elements is given by the combination formula: $$\binom{n}{2} = \frac{n!}{2!(n-2)!}$$ 5. **Apply the formula:** $$\binom{4}{2} = \frac{4!}{2! \times 2!} = \frac{4 \times 3 \times \cancel{2!}}{2! \times \cancel{2!}} = \frac{12}{2} = 6$$ 6. **Answer:** There are 6 subsets of $S$ that contain exactly 2 elements.