Describe an algorithm that, given $n$ integers in the range $0$ to $k$, preprocesses its input and then answers any query about how many of the $n$ integers fall into a range $[a..b]$ in $\O(1)$ time. Your algorithm should use $\Theta(n+k)$ preprocessing time.
This is not even challenging.
We just take the part of
COUNTING-SORT that builds up the array
we want to count the number of integers in $[a..b]$, we take
C[b] - C[a-1]
C[-1] = 0). This yields the number of integers in the given range.