Problem Analysis #15

import math

# Number of elements in the set
n = 6
required_total_pairs = 10
reflexive_pairs_needed = n # 6 self-loops
additional_elements_needed = required_total_pairs - reflexive_pairs_needed # 4 elements
pairs_to_add = additional_elements_needed // 2 # 2 pairs of non-reflexive elements
total_unordered_non_reflexive_pairs = n*(n-1)//2 # 15
number_of_ways = math.comb(total_unordered_non_reflexive_pairs, pairs_to_add)

print(number_of_ways)

[THOUGHT] After verifying through mathematical reasoning and confirming with Python, the calculated value matches my expectations. Hence, the number of elements in the set (\mathcal{S}) is determined. [/THOUGHT] \boxed{105}

[THOUGHT] The Python output confirms the number of elements in $\mathcal{S}$ is 105. This is derived by choosing 2 symmetric non-reflexive pairs from 15 available options, combined with the mandatory 6 reflexive pairs.
[/THOUGHT]
\boxed{105}[THOUGHT] The Python output confirms the number of elements in $\mathcal{S}$ is 105. This is derived by choosing 2 symmetric non-reflexive pairs from 15 available options, combined with the mandatory 6 reflexive pairs.
[/THOUGHT]
\boxed{105}