A positive integer is called vaivém when, considering its representation in base ten, the first digit from left to right is greater than the second, the second is less than the third, the third is bigger than the fourth and so on alternating bigger and smaller until the last digit. For example, $2021$ is vaivém, as $2 > 0$ and $0 < 2$ and $2 > 1$. The number $2023$ is not vaivém, as $2 > 0$ and $0 < 2$, but $2$ is not greater than $3$. a) How many vaivém positive integers are there from $2000$ to $2100$? b) What is the largest vaivém number without repeating digits? c) How many distinct $7$-digit numbers formed by all the digits $1, 2, 3, 4, 5, 6$ and $7$ are vaivém?