Introduction:
The symmetric group S5 consists of all possible permutations of 5 elements. An element in S5 is said to be its own inverse if the permutation, when applied twice, results in the original permutation. In other words, for an element σ in S5, σ × σ = e, where e is the identity permutation. We need to determine the number of elements in S5 that satisfy this condition.

Solution:
To find the number of elements in S5 that are their own inverses, we can consider each possible permutation and check if it satisfies the condition σ × σ = e.

Identity Permutation:
The identity permutation in S5 is the permutation that leaves all elements unchanged. It can be represented as (1)(2)(3)(4)(5), where each number represents the position of the element in the permutation. This permutation satisfies the condition σ × σ = e because applying the identity permutation twice results in the original permutation.

Transpositions:
A transposition is a permutation that swaps two elements while leaving all other elements unchanged. In S5, there are C(5,2) = 10 transpositions. For example, the transposition (12) swaps the positions of elements 1 and 2. These transpositions also satisfy the condition σ × σ = e because applying a transposition twice results in the original permutation.

3-Cycles:
A 3-cycle is a permutation that cyclically permutes three elements while leaving all other elements unchanged. In S5, there are C(5,3) = 10 3-cycles. For example, the 3-cycle (123) cyclically permutes the positions of elements 1, 2, and 3. These 3-cycles also satisfy the condition σ × σ = e because applying a 3-cycle twice results in the original permutation.

4-Cycles:
A 4-cycle is a permutation that cyclically permutes four elements while leaving the fifth element unchanged. In S5, there are C(5,4) = 5 4-cycles. For example, the 4-cycle (1234) cyclically permutes the positions of elements 1, 2, 3, and 4 while leaving element 5 unchanged. These 4-cycles also satisfy the condition σ × σ = e because applying a 4-cycle twice results in the original permutation.

5-Cycles:
A 5-cycle is a permutation that cyclically permutes all five elements. In S5, there is only one 5-cycle, which is represented as (12345). This 5-cycle also satisfies the condition σ × σ = e because applying a 5-cycle twice results in the original permutation.

Counting the Number of Elements:
From the above analysis, we can see that there are 1 identity permutation, 10 transpositions, 10 3-cycles, 5 4-cycles, and 1 5-cycle that satisfy the condition σ × σ = e. Therefore, the total number of elements in S5 that are their own inverses is 1 + 10 + 10 + 5 + 1 = 27.

Conclusion:
The correct answer is option 'A', which is 26. There was an error in the given