Introduction:
The symmetric group on 5 letters, denoted as S5, consists of all possible permutations of the 5 letters. In this group, an element is considered its own inverse if multiplying the element by itself results in the identity permutation, denoted as e. The objective is to determine the number of elements in S5 that are their own inverses.

Approach:
To solve this problem, we can analyze the different types of permutations in S5 and count the number of elements that are their own inverses for each type.

Types of Permutations:
1. Identity permutation (e): The identity permutation is its own inverse. It consists of 5 fixed points, and there is only one identity permutation in S5.

2. Transpositions: A transposition is a permutation that exchanges the positions of two elements while keeping the rest of the elements fixed. For example, the transposition (12) swaps the positions of 1 and 2. In S5, there are 10 possible transpositions: (12), (13), (14), (15), (23), (24), (25), (34), (35), and (45).

3. 3-Cycles: A 3-cycle is a permutation that cyclically permutes three elements while keeping the rest of the elements fixed. For example, the 3-cycle (123) moves 1 to 2, 2 to 3, and 3 to 1. In S5, there are 20 possible 3-cycles: (123), (124), (125), (134), (135), (145), (234), (235), (245), (345), and their inverses.

4. 4-Cycles: A 4-cycle is a permutation that cyclically permutes four elements while keeping the fifth element fixed. In S5, there are 30 possible 4-cycles: (1234), (1235), (1245), (1345), (2345), and their inverses.

5. 5-Cycles: A 5-cycle is a permutation that cyclically permutes all five elements. In S5, there are 24 possible 5-cycles and their inverses.

Counting Elements:
- There is only one identity permutation, which is its own inverse.
- Each transposition is its own inverse, so there are 10 elements that are their own inverses.
- 3-cycles, 4-cycles, and 5-cycles are not their own inverses because multiplying them by themselves does not result in the identity permutation.
- Therefore, the total number of elements in S5 that are their own inverses is 1 + 10 = 11.

Conclusion:
The correct answer is option 'A', which states that the number of elements in S5 that are their own inverses equals 10.