Subject: Mathematics — Permutations and Combinations
Question: Let the set of all relations $R$ on the set $\{ a,b,c,d,e,f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly $10$ elements, be denoted by $S$. Then the number of elements in $S$ is \_\_\_\_\_\_\_\_\_\_.
Answer: $105$
Solution: $(S) = \frac{\binom{6}{2}\binom{4}{2}}{2} + \binom{6}{3}\binom{3}{1}.$
$= 45 + 60 = 105$
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