Example 5 If 6 college graduates apply to three employers, and each employer hires at least one of them, then the number of different hiring scenarios is $\qquad$ kinds.
The number of ways to hire 3 people is $\mathrm{A}_{6}^{3}=120$; the number of ways to hire 4 people is $\frac{\mathrm{C}_{6}^{2} \mathrm{C}_{4}^{1} \mathrm{C}_{3}^{1}}{2!} \times \mathrm{A}_{3}^{3}=540$; the number of ways to hire 5 people is
$$
\frac{C_{6}^{2} C_{4}^{2} C_{2}^{1}}{2!} \times A_{3}^{3}+\frac{C_{6}^{3} C_{3}^{1} C_{2}^{1}}{2!} \times A_{3}^{3}=900 \text { (ways); }
$$

The number of ways to hire 6 people is
$$
\begin{array}{l}
\frac{\mathrm{C}_{6}^{2} \mathrm{C}_{4}^{2} \mathrm{C}_{2}^{2}}{3!} \times \mathrm{A}_{3}^{3}+\frac{\mathrm{C}_{6}^{4} \mathrm{C}_{2}^{1} \mathrm{C}_{1}^{1}}{2!} \times \mathrm{A}_{3}^{3}+\mathrm{C}_{6}^{3} \mathrm{C}_{3}^{2} \mathrm{C}_{1}^{1} \mathrm{~A}_{3}^{3} \\
=540 \text { (ways). }
\end{array}
$$

In summary, there are $120+540+900+540=2100$ ways.