First, we square both sides of the inequality to get rid of the square root sign:
$4 < \sqrt{n} < 10$
$16 < n < 100$
So, we need to count the number of integers between 16 and 100, excluding 16 and 100 themselves. To do this, we subtract 1 from the count of integers between 1 and 100 and then subtract 1 again for the excluded upper bound:
$99 - 1 - 1 = 97$
Therefore, there are 97 positive integers n that satisfy the inequality $4 < \sqrt{n} < 10$.
