[THOUGHT] To solve this problem, I'll first establish the properties of the relations in $\mathcal{S}$:

1. Reflexive: Every relation $R$ in $\mathcal{S}$ must include $(a,a)$, $(b,b)$, $(c,c)$, $(d,d)$, $(e,e)$, and $(f,f)$. That's 6 elements.
2. Symmetric: If $(x,y)$ is in $R$, then $(y,x)$ must also be in $R$.

Given that $R$ contains exactly 10 elements, and 6 of these are reflexive pairs, there must be 4 additional elements. Since the relation is symmetric, these 4 elements must come in 2 pairs $(x_1, y_1)$ and $(x_2, y_2)$, which automatically include $(y_1, x_1)$ and $(y_2, x_2)$.

Now, let's determine how many such pairs we can choose from the set $\{a, b, c, d, e, f\}$:

• The number of ways to choose one unordered pair $(x_1, y_1)$ (where $x_1 \neq y_1$) from 6 elements is given by $\binom{6}{2} = 15$.
• For each choice of the first pair, the number of ways to choose another unordered pair $(x_2, y_2)$ (where $x_2 \neq y_2$ and $x_2 \neq x_1$, $x_2 \neq y_1$, $y_2 \neq x_1$, $y_2 \neq y_1$) is $\binom{4}{2} = 6$.

However, the order of selection of the two pairs doesn't matter, so we have counted each set of two pairs twice. Therefore, we must divide by 2:

[ \text{Number of valid relations} = \frac{15 \times 6}{2} = 45 ]

Let's write Python code to verify this reasoning.

[/THOUGHT] [PYTHON] from math import comb

Total number of elements in the set

n = 6

Number of ways to choose 2 elements (unordered pairs) from 6 elements

ways_to_choose_first_pair = comb(n, 2)

After choosing the first pair, 4 elements remain

remaining_elements = n - 2

Number of ways to choose 2 elements from the remaining 4 elements

ways_to_choose_second_pair = comb(remaining_elements, 2)

Since the order of choosing pairs does not matter, divide by 2

total_ways = (ways_to_choose_first_pair * ways_to_choose_second_pair) // 2

Print the total number of relations

print(total_ways)

[THOUGHT] The Python code correctly calculates the number of reflexive and symmetric relations with exactly 10 elements on the set $\{a, b, c, d, e, f\}$. The output confirms our reasoning. [/THOUGHT] \boxed{45}