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Catherine rolls a standard $6$-sided die six times. If the product of her rolls is $2000,$ then how many different sequences of rolls could there have been? (The order of the rolls matters.)

 Mar 24, 2024
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The prime factorization of 2000 is 24*53.

Catherine must have rolled 3 5s, to form the 53.

The 24 is more tricky. Catherine could've rolled 4's or 2's a certain number of times to form the 24, but the number of rolls to acheive this must be 3, because she rolls a total of 6 rolls.

Consider all the cases:

#1: The rolls 2, 2 and 22 in some order.

#2: The rolls 1 22 and 22 in some order.

These are the only ways she coud've rolled a 4.

The number of ways to arrange 2, 2, 22, 3, 3, 3 is 60

The number of ways to arrange 1, 22, 22, 3, 3, 3 is also 60.

Therefore there could be 60+60 = 120 sequences. 

 Mar 25, 2024

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