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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.)

 Mar 12, 2021
 #1
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The rolls must be 1, 2, 5, 5, 6 in some order, so there are 5!/2 = 60 ways.

 Mar 12, 2021
 #2
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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

 Mar 12, 2021

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