【Example 4】How many four-digit numbers without repeated digits that are divisible by 3 can be formed using the digits $1,2,3,4,5,6$?
We know that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. Since
$$
1+2+3+4+5+6=21
$$

to get a four-digit number that is divisible by 3, the sum of the two digits not used in the number must be divisible by 3. There are 5 such pairs of digits: $1+2=3$, $1+5=6$, $2+4=6$, $3+6=9$, $4+5=9$. Therefore, there are 5 groups of digits that can be used to form the number. Since the digits can be permuted in any order, the total number of four-digit numbers that can be formed without repeating digits and that are divisible by 3 is
$$
5 \cdot 4!=120
$$