1. **State the problem:** We want to find the number of distinct positive integers $n$ such that the product $(n-1)(n-9)(n-17)$ is less than 0. 2. **Understand the expression:** The product $(n-1)(n-9)(n-17)$ changes sign at the roots $n=1$, $n=9$, and $n=17$. 3. **Sign analysis:** The product is zero at $n=1,9,17$. Between these points, the sign of the product changes depending on the number of negative factors. 4. **Intervals to check:** - For $n<1$, all three factors are negative or zero. - For $10$, $(n-9)<0$, $(n-17)<0$ so product is positive (since two negatives make a positive). - For $90$, $(n-9)>0$, $(n-17)<0$ so product is negative (one negative factor). - For $n>17$, all factors are positive, so product is positive. 5. **Find positive integers where product < 0:** From above, product is negative only when $9