Catherine rolls a $6$-sided die five times, and the product of her rolls is $300.$ How many different sequences of rolls could there have been? (The order of the rolls matters.)
The rolls must be 1, 2, 5, 5, 6 in some order, so there are 5!/2 = 60 ways.
There are more ways though. For example, there could be:
1, 3, 4, 5, 5
2, 2, 3, 5, 5
This gives another 90 ways:
1, 3, 4, 5, 5: 5!/2 = 60
2, 3, 3, 5, 5: 5!/4 = 30
If you add it with the previous one, we get 60+60+30 = 150
