Question

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.)

Answer 1

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.