Counting

 

Catherine rolls a standard $6$-sided die six times.  If the product of her rolls is $100,$ then how many different sequences of rolls could there have been?  (The order of the rolls matters.)   

 

The prime factorization of 100 is 2 • 2 • 5 • 5  

 

There are other factors of 100, but you can't use any of them because  

they're larger than 6 and none of the faces on the die is larger than 6.  

 

For a product of 100 with 6 rolls, the numbers have to be 2, 2, 5, 5, 1, 1  

in some order, and the problem stipulates the order of the rolls matters. 

 

For the 1^st roll, there are 6 possibilities   

For the 2^nd roll, there are 5 possibilities  

For the 3^rd roll, there are 4 possibilities  

and so on.  

 

So the total number of possible sequences is 6 • 5 • 4 • 3 • 2 • 1  =  720  

 

OR the numbers could be 4, 5, 5, 1, 1, 1  

in which case it's still 6! to roll those so it's 720 ways to get 100 that way, too. 

 

Mind that this is not a probability, this is just the number of ways to get 100.  

_.