Find the missing number?

The answer is $\boxed{8}$.

The missing number is the sum of the first three numbers in the sequence, $1$, $2$, and $3$, plus the product of the first two numbers, $1$ and $2$. This is because the sequence is generated by the formula $a_n = (n-1)a_{n-2} + (n-2)a_{n-3}$, where $a_n$ is the $n$th term in the sequence and $a_0 = 1$ and $a_1 = 2$.

Here is a brief explanation of each option:

• Option A: $3$ is not the sum of the first three numbers in the sequence, $1$, $2$, and $3$.
• Option B: $8$ is the sum of the first three numbers in the sequence, $1$, $2$, and $3$, plus the product of the first two numbers, $1$ and $2$.
• Option C: $10$ is not the sum of the first three numbers in the sequence, $1$, $2$, and $3$, plus the product of the first two numbers, $1$ and $2$.
• Option D: $4$ is not the sum of the first three numbers in the sequence, $1$, $2$, and $3$, plus the product of the first two numbers, $1$ and $2$.