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Subject: Need help verifying permutation math rss

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This Guy
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I'm working on a 2 player game about taking resources, each of which has 4 aspects with 2-3 values each: 123+abc+pdq+xy. So one resource might have the aspects 2bdx. This yields 54 unique resources. Players draw one resource to represent themselves during the game, leaving 52 resources up for grabs throughout the game. This is all good.

When trying to take a resource, I used to give you a bonus based on how many aspects you had in common with the resource: which meant you would get +0 to +3. When you won the resources, its point value was equal to the number of matches the resource had with your opponent. This calculation seemed to be too chunky, and some other nice mechanisms I've put in place have made bonuses too high.

Instead of a bonus, I am considering making it so you get a malus to your action which is equal to the number of misses you have with the resource, which is also the point value of the resource. So a resource with which you have no matches would give you -4 to your grab, but be worth 4 points if you succeed.

I need help verifying the number of maluses this system would create. Manually trying to do this, I came up with:
07 one point resources
17 two point resources
20 three point resources
09 four point resources
(you and your opponent should both knock the same point value out of each other's pool)

I'd appreciate somebody posting the math, as I'm incredibly deficient there.

Thanks in advance.
 
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Matt Davis
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I count:

7 1-pointers
18 2-pointers
20 3-pointers
8 4-pointers

To count these, you can think about what choices you have to make in building one of those things - and just count how many options you had along the way.

To count the 1-pointers - each resource has 4 aspects, leaving 7 it doesn't have. To pick a 1-pointer, you just pick one of the aspects it doesn't have and switch its corresponding aspect to that - 7 ways to do that.

To count the 2-pointers - There are two cases here. Maybe it matches the xy aspect and one of the others. Then you have to choose which other one it matches - 3 ways to do that. And once you do that, you have 2 choices each for what the non-matching aspects could be. So that's 3 x 2 x 2 = 12. If it matches two of the 3-value aspects, it doesn't match the xy - but we have no choices to make there. So all we choose is which 3-value aspect it doesn't match, and which of the two values it is. That's 3 x 2 = 6, for a total of 18.

To count the 3-pointers - If it matches the xy-aspect, then we get two choices each for the values of the other aspect, which is 2 x 2 x 2 = 8 ways. If it matches one of the 3-value aspects, we get 3 choices of which one, and then we have to choose which of the two other values the other two non-matching aspects should have. (Again, if the xy aspect doesn't match, we have no choice.) So that's 3 x 2 x 2 = 12, for a total of 20.

To count the 4-pointers - Nothing matches, so we get to choose the other values for the 3-value aspects - we have 2 choices for each of those. So that's 2 x 2 x 2 = 8 resources.
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Christopher
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Matt is correct , and faster than me . (I just finished my prep-work on a piece of paper and wanted to upload it, when I saw his answer.)
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This Guy
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Thanks, gang.
 
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