Problem Analysis #15

[THOUGHT] We need to count reflexive, symmetric relations on a 6-element set with exactly 10 ordered pairs. A reflexive relation on 6 elements must include all 6 diagonal pairs. Symmetry forces off-diagonal pairs to come in 2-element symmetric pairs ((x,y),(y,x)). So if there are k off-diagonal ordered pairs, k must be even, and total size is 6 + k = 10, giving k = 4. Thus we need exactly 4 off-diagonal ordered pairs, i.e., 2 unordered pairs from the 15 possible unordered pairs of distinct elements. So the answer should be C(15,2). I’ll compute that with Python. [/THOUGHT] [PYTHON] import math

result = math.comb(15, 2) print(result) [/PYTHON]

[THOUGHT] Using the Python-verified result for C(15,2), the number of such reflexive, symmetric relations with exactly 10 elements is 105. $\boxed{105}$