Example 3 The integer array $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ satisfies the condition $0<x_{1} \leqslant x_{2} \leqslant x_{3} \leqslant x_{4}<$ 7 , how many such arrays are there?
Solution: Make the transformation $y_{i}=x_{i}-1(i=1,2,3,4)$, then $0<x_{1} \leqslant x_{2} \leqslant x_{3} \leqslant x_{4}<7$ is transformed into $-1<y_{1} \leqslant y_{2} \leqslant y_{3} \leqslant y_{4}<6$, that is, the ordered array $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ is transformed into a non-negative ordered integer array $\left(y_{1}, y_{2}, y_{3}, y_{4}\right)$.

Construct the following model: Take 5 identical balls and 4 separators $I$, and arrange them in a row to occupy 9 positions. This can be done as follows: First arrange 4 separators $I$ in 9 positions, with a total of $C_{9}^{4}$ arrangements, then place 5 identical balls in the remaining positions, so that each arrangement corresponds one-to-one to a non-negative ordered integer array $\left(y_{1}, y_{2}, y_{3}, y_{4}\right)$, for example,
$I 00 I 0 I 00 I \leftrightarrow(0,2,3,5)$;
$0 I 00 I I 0 I 0 \leftrightarrow(1,3,3,4)$, etc.
By the transformation made, $\left(y_{1}, y_{2}, y_{3}, y_{4}\right)$ corresponds one-to-one to $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$, hence the number of arrays sought is $C_{9}^{4}=126$.