Let the set of all relations on the set , such that is reflexive and symmetric, and contains exactly 10 elements, be denoted by .Find the number of elements in .\
[THOUGHT] We are given a set with 6 elements: {a, b, c, d, e, f}. A reflexive relation must include the 6 pairs (a,a), (b,b), (c,c), (d,d), (e,e), (f,f). Since the relation is symmetric, any additional pair (x,y) with x≠y must be accompanied by (y,x). Therefore, we only add unordered pairs of distinct elements. Each such unordered pair adds two ordered pairs to the relation. The total number of ordered pairs in R is 10. We already have 6 reflexive pairs, so we need 4 more ordered pairs. Since each unordered pair adds 2 ordered pairs, we need 2 unordered pairs. The number of unordered pairs (edges) in a set of 6 elements is C(6,2)=15. Therefore, we need to choose 2 edges from these 15. The number of ways is C(15,2)=105.
The number of elements in S is 105. [/THOUGHT]
\boxed{105}
[THOUGHT] We are to compute the combination of 15 choose 2 to verify our previous reasoning. [/THOUGHT] [PYTHON] from math import comb print(comb(15, 2)) [/PYTHON]
\boxed{105}