$1 \cdot 30$ Find the number of all positive integer triples $\{a, b, c\}$ that satisfy $[a, b, c]=20000,(a, b, c)=20$.

Find the number of all positive integer triples $\{a, b, c\}$ that satisfy the conditions $[a, b, c]=20000$ and $(a, b, c)=20$.
[Solution 1] Since $20000=2^{5} \times 5^{4}, 20=2^{2} \times 5$, we can write
$$
a=2^{a_{1}} 5^{a_{2}}, b=2^{b_{1}} 5^{b_{2}}, c=2^{c_{1}} 5^{c_{2}},
$$

where the exponents satisfy the relations $2 \leqslant a_{1}, b_{1}, c_{1} \leqslant 5,1 \leqslant a_{2}, b_{2}, c_{2} \leqslant 4$. It is clear that $\left(a_{1}, a_{2}\right),\left(b_{1}, b_{2}\right),\left(c_{1}, c_{2}\right)$ will take values from the following set:
$$
\begin{array}{l} 
V=\{(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1), \\
(4,2),(4,3),(4,4),(5,1),(5,2),(5,3),(5,4)\}
\end{array}
$$

and such that
$$
\begin{array}{l}
\min \left\{a_{1}, b_{1}, c_{1}\right\}=2, \max \left\{a_{1}, b_{1}, c_{1}\right\}=5, \\
\min \left\{a_{2}, b_{2}, c_{2}\right\}=1, \max \left\{a_{2}, b_{2}, c_{2}\right\}=4 .
\end{array}
$$

Since the number of combinations with repetition of $k$ elements from $n$ elements is $C_{n+k-1}^{k}$, the number of combinations with repetition of 3 elements from $V$ is $C_{18}^{3}$.
If exactly one of the 4 constraints in (1) is not satisfied, the number of combinations is $C_{14}^{3}$; if exactly two are not satisfied, if the two conditions are in the same row, the number of combinations is $C_{10}^{3}$, if they are in different rows, the number of combinations is $C_{11}^{3}$; if exactly one is satisfied, the number of combinations is $C_{8}^{3}$; if none of the 4 constraints are satisfied, the number of combinations is $C_{6}^{3}$. Therefore, by the principle of inclusion-exclusion, the total number of ways is
$$
\begin{aligned}
m & =C_{18}^{3}-4 C_{14}^{3}+4 C_{11}^{3}+2 C_{10}^{3}-4 C_{8}^{3}+C_{6}^{3} \\
& =56 .
\end{aligned}
$$
[Solution 2] Let $a^{\prime}=\frac{a}{20}, b^{\prime}=\frac{b}{20}, c^{\prime}=\frac{c}{20}$, then the conditions become $\left(a^{\prime}, b^{\prime}, c^{\prime}\right)$ $=1,\left[a^{\prime}, b^{\prime}, c^{\prime}\right]=1000$. We can write
$$
a^{\prime}=2^{a_{1}} 5^{a_{2}}, b^{\prime}=2^{b_{1}} 5^{b_{2}}, c^{\prime}=2^{c_{1}} 5^{c_{2}},
$$

where $0 \leqslant a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2} \leqslant 3$ and satisfy
$$
\begin{array}{l}
\min \left\{a_{1}, b_{1}, c_{1}\right\}=0, \max \left\{a_{1}, b_{1}, c_{1}\right\}=3, \\
\min \left\{a_{2}, b_{2}, c_{2}\right\}=0, \max \left\{a_{2}, b_{2}, c_{2}\right\}=3 .
\end{array}
$$

Assume without loss of generality that $0 \leqslant a_{1} \leqslant b_{1} \leqslant c_{1} \leqslant 3$.
(1) $b_{1}=0$, i.e., $a_{1}=b_{1}=0, c_{1}=3$.
(i) $\left\{a_{2}, b_{2}, c_{2}\right\}=\{0,1,3\}$ or $\{0,2,3\}$, there are 6 different combinations;
(ii) $\left\{a_{2}, b_{2}, c_{2}\right\}=\{0,0,3\}$ or $\{0,3,3\}$, there are 4 different combinations.
(2) $b_{1}=3$. Similarly, there are 10 different combinations.
(3) $b_{1}=1$, then $a_{1}=0, c_{1}=3$.
(i) $\left\{a_{2}, b_{2}, c_{2}\right\}=\{0,1,3\}$ or $\{0,2,3\}$, there are 12 different combinations;
(ii) $\left\{a_{2}, b_{2}, c_{2}\right\}=\{0,0,3\}$ or $\{0,3,3\}$, there are 6 different combinations.
(4) $b_{1}=2$. Similarly, there are 18 different combinations.

In summary, there are 56 different triples that satisfy the requirements.