2020 AMC 12B Problem 15

Label the people $0$ through $9.$ Allowed pairings use neighbor edges $(i, i + 1)$ or diameter edges $(i, i + 5).$ Count perfect matchings by the number of diameter edges used. Using no diameters, the ten people split into adjacent pairs in $2$ ways (all “even” edges or all “odd” edges). Using exactly one diameter, choose it in $5$ ways; the remaining two arcs of four people each pair up uniquely, giving $5.$ Using all five diameters gives $1$ matching. If at least one diameter is used, following the forced adjacent pairings around the circle shows that the number of diameters must be odd. With exactly three diameters, the four remaining people must be two opposite adjacent pairs. The position of one such opposite pair determines the matching, giving $5$ possibilities. Hence the total is $2+5+5+1=13.$ Thus, the correct answer is C.