Example 4.27 Using the digits $1,2,3,4$ to form 6-digit numbers, each digit may not appear more than 2 times in the 6-digit number. How many different 6-digit numbers can be formed?
Let the required value be $e$, then $e$ is the coefficient of $t^{6} / 6!$ in the expansion of
$$
E(t)=\left(1+t+t^{2} / 2!\right)^{4}
$$

Expanding $E(t)$, we have
$$
\begin{aligned}
E(t)= & \frac{1}{16}\left[t^{2}+(2 t+2)\right]^{4} \\
= & \frac{1}{16}\left[t^{8}+4 t^{6}(2 t+2)+6 t^{4}(2 t+2)^{2}+\right. \\
& \left.4 t^{2}(2 t+2)^{3}+(2 t+2)^{4}\right],
\end{aligned}
$$

Therefore,
$$
e=\frac{6!}{16}\left(4 \times 2+6 \times 2^{2}\right)=1440 .
$$