Question 61
Mathlandia has $2022$ cities. Show that the number of ways to construct $2021$ roads connecting pairs of cities such that it is possible to get between any two cities, there are no loops, and each city has exactly one or three roads coming out of it is given by
$$
\frac{2022 ! \cdot 2019 !!}{1012 !}
$$
(The notation $2019!!$ means $2019 \cdot 2017 \cdot \cdots \cdot 3 \cdot 1$.)
$$
\frac{2022 ! \cdot 2019 !!}{1012 !}
$$
(The notation $2019!!$ means $2019 \cdot 2017 \cdot \cdots \cdot 3 \cdot 1$.)