Introduction:
We are given that S3 is the group of all permutations on three symbols, and we need to find the number of elements in S3 that satisfy the equation x^2 = e, where e is the identity element.

Explanation:
To solve this problem, we need to find the elements in S3 that, when squared, give the identity element. Let's break it down step by step.

Step 1: Finding all permutations in S3:
To find all permutations in S3, we consider all possible arrangements of the three symbols: 1, 2, and 3. These arrangements can be represented as permutations.

Step 2: Identifying the identity element:
The identity element in a group is the element that, when combined with any other element, leaves that element unchanged. In S3, the identity element can be represented as (1)(2)(3), where each number in parentheses represents the position of the symbol.

Step 3: Squaring the elements:
To find the elements that satisfy the equation x^2 = e, we need to square each element in S3 and check if it equals the identity element (e).

Step 4: Squaring the elements:
Let's list all the elements in S3 and square each one to check if it satisfies x^2 = e:

1. (1)(2)(3): Squaring this element gives (1)(2)(3)(1)(2)(3) = (1)(2)(3), which is the identity element (e).
2. (1)(3)(2): Squaring this element gives (1)(3)(2)(1)(3)(2) = (1)(2)(3), which is the identity element (e).
3. (2)(1)(3): Squaring this element gives (2)(1)(3)(2)(1)(3) = (1)(3)(2), which is not the identity element (e).
4. (2)(3)(1): Squaring this element gives (2)(3)(1)(2)(3)(1) = (1)(2)(3), which is the identity element (e).
5. (3)(1)(2): Squaring this element gives (3)(1)(2)(3)(1)(2) = (2)(3)(1), which is not the identity element (e).
6. (3)(2)(1): Squaring this element gives (3)(2)(1)(3)(2)(1) = (2)(1)(3), which is not the identity element (e).

Step 5: Counting the elements:
From the above calculations, we can see that there are 4 elements in S3 that satisfy the equation x^2 = e.

Conclusion:
Therefore, the number of elements in S3 that satisfy the equation x^2 = e is 4.