imc

Since we are asked for the number of positive $4$-digit integers with at least $2$ or $3$ in it, we can find this by finding the total number of $4$-digit integers and subtracting off those which do not have any $2$s or $3$s as digits. The total number of $4$-digit integers is $9\cdot 10\cdot 10\cdot 10=9000$, since we have $10$ choices for each digit except the first (which can't be $0$). Similarly, the total number of $4$-digit integers without any $2$ or $3$ is $7\cdot 8\cdot 8\cdot 8=3584$. Therefore, the total number of positive $4$-digit integers that have at least one $2$ or $3$ is $9000-3584=\boxed{5416}.$