Catherine rolls a standard 6-sided die five times, and the product of her rolls is 100. How many different sequences of rolls could there have been? (The order of the rolls matters.)
Prime factorizing 100 yields $100=2^2\cdot5^2$. We can think of rolling a 6-sided die as choosing one of the numbers $\{1,2,3,4,5,6\}$, each of which has a different prime factorization. Thus, in order for the product of 5 rolls to be 100, we must choose two 2's and two 5's, and any other choice of the remaining roll. There are $\binom{5}{2}=10$ ways to choose which rolls will be 2's and which will be 5's, and there are $4$ numbers not equal to 2 or 5, so there are $4$ choices for the remaining roll. Therefore, there are $10\cdot4=\boxed{40}$ possible sequences of rolls.
