non-standard dice - ways

I have 7 dice
each die has face values of {1,1,1,5,10,25}
They are my coin dice and I roll them.

How many ways for each sequence below?

How to easily figure it.

1) 1,1,1,1,1,1,1
2) 5,1,1,1,1,1,1
3) 5,5,1,1,1,1,1
4) 10,5,1,1,1,1,1 (I counted 10206 ways for this sequence)


I can't do it
I have to count the ways

thank you
Sally

For the last one, 3^5 x 7 x 6 = 10,206.

Quote: mustangsally

How to easily figure it.

1) 1,1,1,1,1,1,1
2) 5,1,1,1,1,1,1
3) 5,5,1,1,1,1,1
4) 10,5,1,1,1,1,1 (I counted 10206 ways for this sequence)

If I am understanding the problem correctly:

First, if you have N numbers, and N₁ of them are one number, N₂ are a second number, ..., and N_k are a kth number, then the number of ways they can appear is N! / (N₁! x N₂! x ... x N_k!).

For #1, there are 7 1s, so they can appear once; on each die, 1 appears 3 times, so the total is 1 x 3⁷ = 2187.

For #2, there are 7! / (1! 6!) = 7 ways for the numbers to appear; the one 5 appears once on that die, and each of the six 1s appear 3 times on each of those dice, so the total is 7 x 1 x 3⁶ = 5103 ways.

For #3, there are 7! / (2! 5!) = 21 ways for the numbers to appear; each of the two 5s appears once on each of those dice, and each of the five 1s appears three times on each, so the total is 21 x 1² x 3⁵ = 5103 ways.

For #4, there are 7! / (1! 1! 5!) = 42 ways for the numbers to appear; the 10 appears once on that die, the 5 appears once on that die, and each of the four 1s appears three times on each, so the total is 42 x 1 x 1 x 3⁵ = 10,206.

Here's one for you: 25, 10, 10, 5, 5, 1, 1

Quote: ThatDonGuy

For #2, there are 7! / (1! 6!) = 7 ways for the numbers to appear; the one 5 appears once on that die, and each of the six 1s appear 3 times on each of those dice, so the total is 7 x 1 x 3⁶ = 5103 ways.

For #3, there are 7! / (2! 5!) = 21 ways for the numbers to appear; each of the two 5s appears once on each of those dice, and each of the five 1s appears three times on each, so the total is 21 x 1² x 3⁵ = 5103 ways.

#2 and #3 got me.
same # of ways and I stopped.

I left out one important step after that.
thank you so much
easy when one sees how
Sally