1. At a concert, there are 20 singers who will perform. For each singer, there is a set of other singers (possibly an empty set) such that he wishes to perform later than all the singers in this set. Question: Is there a way to have exactly 2,010 ways to order the singers so that all their wishes are satisfied?
1. Such examples exist.

A ranking of singers that satisfies everyone's wishes is called "good".

If for a set of wishes of $k$ singers there exist $N$ good rankings, then $N$ is called "achievable by $k$ singers" (or simply "$k$-achievable").
Next, we prove that 2010 is "20-achievable".
First, we prove a lemma.
Lemma Assume $n_{1}$ and $n_{2}$ are $k_{1}$-achievable and $k_{2}$-achievable, respectively. Then $n_{1} n_{2}$ is $\left(k_{1}+k_{2}\right)$-achievable.

Proof Suppose singers $A_{1}, A_{2}, \cdots, A_{k_{1}}$ (some of whom have wishes) can achieve $n_{1}$, and singers $B_{1}, B_{2}, \cdots, B_{k_{2}}$ (some of whom have wishes) can achieve $n_{2}$, and add the wish for each singer $B_{i}$: to perform later than all singers $A_{j}$. Then the good rankings of singers have the form
$$
\left(A_{i_{1}}, A_{i_{2}}, \cdots, A_{i_{k_{1}}}, B_{j_{1}}, B_{j_{2}}, \cdots, B_{j_{k_{2}}}\right),
$$
where, ( $\left.A_{i_{1}}, A_{i_{2}}, \cdots, A_{i_{k_{1}}}\right)$ is a good ranking for $A_{i}$, and $\left(B_{j_{1}}, B_{j_{2}}, \cdots, B_{j_{k_{2}}}\right)$ is a good ranking for $B_{j}$.

Conversely, every ranking of this form is clearly good, so the number of good rankings is $n_{1} n_{2}$.
Back to the original problem.
By the lemma, construct the number of achievable rankings for 4 singers, 3 singers, and 13 singers as $5, 6, 67$, respectively. Then
$$
2010=5 \times 6 \times 67
$$

is achievable by $4+3+13=20$ singers.
Examples of these three scenarios are shown in Figures 1 to 3 (the numbers in parentheses are the number of good rankings, and wishes are indicated by arrows).

In Figure 1, $c$ wishes to perform later than $a$ and $b$, and $d$ wishes to perform later than $b$. Then there are exactly 5 good rankings
$$
\begin{array}{l}
(a, b, c, d), (a, b, d, c), (b, a, c, d), \\
(b, a, d, c), (b, d, a, c).
\end{array}
$$

In Figure 2, every ranking is good, totaling 6, as they have no wishes.
In Figure 3, the order of $a_{1}, a_{2}, \cdots, a_{11}$ is fixed on this line, singer $x$ can be after each $a_{i}(i \leqslant 9)$, singer $y$ can be before each $a_{j}(j \geqslant 5)$, and when between $a_{i}$ and $a_{i+1}(5 \leqslant i \leqslant 8)$, the order of the two singers can be swapped.
Thus, the number of good rankings is
$$
9 \times 7+4=67.
$$