Number of ways of making change in Big O notation
Suppose that the two denominations are $a$ and $b$. Every solution to the coin-changing problem is a solution to the equation $\alpha a + \beta b = n$ in non-negative integers $\alpha,\beta$. For every possible value of $\alpha$ there is at most one solution. Since $0 \leq \alpha \leq n/a$, we conclude that there are at most $n/a + 1 = O(n)$ solutions.