Plz help me

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.)

sandwich Oct 28, 2023

#1**0 **

*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.

_{.}

Bosco Oct 28, 2023