How many $4$ digit numbers divisible by $4$ can be formed using the digits $0,1,2,3,4$ (without repetition)?

Question

Here is my approach:- Firstly, I fixed the last digit as $4$ then there will be only $2$ numbers $(0,2)$ for the ten's digit, $3$ numbers for the hundred's digit and $2$ numbers for the Thousand's digit (so that they don't repeat). Number of $4$ digit numbers in which $4$ is the last digit and is divisible by $4 = 2 \times 3 \times 2 = 12$. As there can be only $4,2,0$ as the last digit so there are $12\times 3 = 36$ numbers possible but that is an incorrect answer. The correct answer is $30$. Where did I go wrong?

Answer 1

Case 1: $\underbrace{**}_{\{1,3,4\}}20 \Rightarrow P(3,2)=3!=6$.

Case 2: $**04 \Rightarrow P(3,2)=3!=6$.

Case 3: $**40 \Rightarrow P(3,2)=3!=6$.

Case 4: $**12=\underbrace{**}_{\{3,4\}}12+\underbrace{*}_{\{3,4\}}012 \Rightarrow P(2,2)+C(2,1)=4$.

Case 5: $**32 \Rightarrow P(2,2)+C(2,1)=4$.

Case 6: $**24 \Rightarrow P(2,2)+C(2,1)=4$.