Question 6
An integer is called *snakelike* if its decimal representation $a_1 a_2 a_3 \ldots a_k$ satisfies $a_i \lt a_{i+1}$ if $i$ is odd and $a_i \gt a_{i+1}$ if $i$ is even. How many snakelike integers between $1000$ and $9999$ have four distinct digits?