Catherine rolls a standard 6-sided die six times. If the product of her rolls is \(2500\) then how many different sequences of rolls could there have been? (The order of the rolls matters.)

Guest Jun 21, 2023

#2**0 **

*Catherine rolls a standard 6-sided die six times. If the product of her rolls is 2500 then how many different sequences of rolls could there have been? (The order of the rolls matters.) *

There are two ways to obtain a product of 2500 using factors between 1 and 6 inclusive.

One way is 5 • 5 • 5 • 5 • 2 • 2

The other way is 5 • 5 • 5 • 5 • 4 • 1

I admit I had to look that up.

I searched on google for prime factorization of 2500.

I don't know how to use permutations to mix those other numbers

in there with all those fives, so I would have to use brute force.

225555 522555 552255 555225 555522

252555 525255 552525 555252

255255 525525 552552

255525 525552

255552

If I didn't miss any, there are 15 with 5's and 2's.

There would be another 30 with 5's, a 4, and a 1.

This is because there would be 15 with the 1 first,

then another 15 with the 4 first. I think that's right.

I'm not going to write them all down. If you want to

write them all down, that would be okay with me.

I think your teacher would be impressed. ;-)

Anyway, the total number of sequences,

according to my feeble calculations, seems to be **45 sequences**.

_{.}

Bosco Jun 22, 2023

#3**-7 **

**Hi Ron,**

\(\small \text {Here's the formula to solve for unique distinguishable permutation counts where the set of elements has one or more subsets of identical elements }\\ \small \text {(Each redundancy is factored out.) }\\ \; \dfrac {^n \mathrm P_r} {\mathrm d_1! \cdot \mathrm d_2! \; \cdots \mathrm d_x! } | \text { Where } (\mathrm d_x) \text{ is the number of duplicates for each subset of duplicates.}\\ \small \text {Because this solution does not require subset permutation counts, } (^n \mathrm P_r) \text { simplifies to } \mathrm (\mathrm P!) \\ \dfrac {\mathrm P!} {\mathrm d_1! \cdot \mathrm d_2! \; \cdots \mathrm d_x! }\Rightarrow \dfrac{6!}{4! * 2!} = 15 \; \; \text{ and } \; \dfrac{6!}{4! } = 30 \)

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It’s wonderful that you’ve become a member, Ron. I’m sure your cat is pleased with your user name.

GA

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GingerAle
Jun 22, 2023

#4**0 **

Hey Ginger!

It's so great to hear from you. I haven't been seeing postings of yours for quite a while, and I was afraid I never would again. Thank you for that explanation about permutations. I agree that Bosco the Chocolate Bear would have been pleased with my username. Or is pleased, if he's watching from cat heaven.

This is something funny. It turns out that my username here is a cat I used to have, and my user name on eBay is a cat my grandma used to have. I didn't plan it that way, and the coincidence just dawned on me a couple of days ago.

Ron

Bosco
Jun 23, 2023