$1 \cdot 29$ Given $\lg 2=0.30103$, calculate how many digits $M=1+10^{4}+\frac{10^{4}\left(10^{4}-1\right)}{1 \cdot 2}+$ $\frac{10^{4}\left(10^{4}-1\right)\left(10^{4}-2\right)}{1 \cdot 2 \cdot 3}+\cdots+\frac{10^{4}\left(10^{4}-1\right)}{1 \cdot 2}+10^{4}+1$ has.
[Solution]
$$
\begin{aligned}
M= & (1+1)^{10^{4}}=2^{10000} \\
\lg M & =10000 \lg 2 \\
& =10000 \times 0.30103 \\
& =3010.3
\end{aligned}
$$

Therefore, $M$ has 3011 digits.