Example 1-21 Laying a $1 \times 7$ module with $1 \times 1, 1 \times 2, 1 \times 3$ blocks, how many patterns are there?
(1) Using 7 pieces of $1 \times 1$ finished products, the pattern is 1 type;
(2) Using 5 pieces of $1 \times 1$ and 1 piece of $1 \times 2$ finished products, the pattern is $6!/ 5!=6$ types;
(3) Using 4 pieces of $1 \times 1$ and 1 piece of $1 \times 3$ finished products, the pattern is $5!/ 4!=5$ types;
(4) Using 3 pieces of $1 \times 1$ and 2 pieces of $2 \times 2$ finished products, the pattern is $\frac{5!}{3!2!}=10$ types;
(5) Using 2 pieces of $1 \times 1$, 1 piece of $1 \times 2$, and 1 piece of $1 \times 3$ finished products, the pattern is $4!/ 2!=12$ types;
(6) Using 1 piece of $1 \times 1$ and 3 pieces of $1 \times 2$ finished products, the pattern is $4!/ 3!=4$ types;
(7) Using 1 piece of $1 \times 1$ and 2 pieces of $1 \times 3$ finished products, the pattern is $3!/ 2!=3$ types;
(8) Using 2 pieces of $1 \times 2$ and 1 piece of $1 \times 3$ finished products, the pattern is $3!/ 2!=3$ types.
According to the addition rule, there are a total of
$1+6+5+10+12+4+3+3=44$ types of patterns.