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# Counting

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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.)

Oct 28, 2023

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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 1st roll, there are 6 possibilities

For the 2nd roll, there are 5 possibilities

For the 3rd 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.

.

Oct 28, 2023