2021 Fall AMC 10B Problem 23

Count the complement: colorings of the $10$ edges of $K_5$ with no monochromatic triangle. At any vertex, if $3$ incident edges had the same color, then the edges among their other endpoints would all have to be the other color, making a monochromatic triangle. Thus each vertex has exactly $2$ red and $2$ blue incident edges. So the red edges form a $2$-regular graph on $5$ vertices, which must be a $5$-cycle. The number of labeled $5$-cycles is $\frac{(5-1)!}{2}=12.$ There are $2^{10}=1024$ total colorings, so the desired probability is $$1-\frac{12}{1024}=\frac{253}{256}.$$ Thus, the answer is D .