Example 1.9 How many 5-digit numbers greater than 35000 can be formed using $1,2,3,4,5,6$?
Solution: Let the number of 5-digit numbers greater than 35000 formed by $1,2,3,4,5,6$ be $N$. Then these $N$ 5-digit numbers can be divided into the following two categories:
(1) 5-digit numbers with the ten-thousands digit being 3.
For 5-digit numbers in this category, the thousand's digit must be 5 or 6, so there are $2 \times 6^{3}=632$ such 5-digit numbers.
(2) 5-digit numbers with the ten-thousands digit greater than 3.
For 5-digit numbers in this category, the ten-thousands digit must be 4, 5, or 6, so there are $3 \times 6^{4}=3888$ such 5-digit numbers.
By the addition principle, we get
$$
N=632+3888=4520 .
$$