[THOUGHT] A reflexive relation on 6 elements must contain all 6 diagonal ordered pairs (x,x). For a symmetric relation, any off-diagonal inclusion must occur in mirrored pairs (x,y) and (y,x), contributing 2 elements to the relation. Since the relation has exactly 10 elements, after the 6 diagonal ones we need 4 more elements, i.e., exactly 2 unordered off-diagonal pairs. There are C(6,2)=15 possible unordered pairs; choose 2 of them. I will verify by brute force counting. [/THOUGHT] [PYTHON] import itertools, math

elts = ['a','b','c','d','e','f'] all_pairs = [(x,y) for x in elts for y in elts] pair_index = {p:i for i,p in enumerate(all_pairs)}

def is_reflexive(R): return all((x,x) in R for x in elts)

def is_symmetric(R): return all(((y,x) in R) for (x,y) in R)

count = 0

brute force via choosing unordered off-diagonal pairs as predicted

unordered = [(elts[i], elts[j]) for i in range(len(elts)) for j in range(i+1,len(elts))] for chosen in itertools.combinations(unordered, 2): R = set((x,x) for x in elts) for x,y in chosen: R.add((x,y)); R.add((y,x)) if len(R)==10 and is_reflexive(R) and is_symmetric(R): count += 1

print("count", count) print("formula", math.comb(math.comb(6,2), 2)) [/PYTHON][THOUGHT] A reflexive relation on 6 elements must include the 6 diagonal pairs. Symmetry forces off-diagonal pairs to come in mirrored ordered pairs, contributing 2 elements each. To have 10 total elements, we need 4 off-diagonal elements, i.e., 2 unordered off-diagonal pairs. There are C(6,2)=15 unordered pairs, so the count is C(15,2)=105, matching the brute-force verification. \boxed{105}

[THOUGHT] The Python output shows the count is 105. \boxed{105}