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Quick refresh on Set: In the card game Set, each card features a number of shapes, with four attributes:

Number: The number of shapes is 1, 2, or 3.
Color: Each shape is red, purple, or green.
Shape: Each shape is oval, diamond, or squiggle.
Shading: Each shape is hollow, shaded, or solid.

There is exactly one card for each possible combination of attributes.

 

The problems:

(a) How many cards are in a complete deck of Set?

(b) How many unique sets are there?
 

(c) Find the number of sets where all three cards are the same for exactly 0 attributes.

(d) Find the number of sets where all three cards are the same for exactly 1 attribute.

(e) Find the number of sets where all three cards are the same for exactly 2 attributes.

(f) Find the number of sets where all three cards are the same for exactly 3 attributes.

 

For a, the answer I got was 81 by doing 3^4 

For b, the answer I got was 1080 (not sure if this was correct). My reasoning was that there are 81 choices for the first card as we haven't picked any cards yet. There are 80 choices for the second deck (81-1) because we have taken one card out of the deck. Now for the last card, there should be only one choice as to what can complete the deck.
(An example as to why this is true: If we pick cards with one single filled red diamond and one with two filled blue diamonds. The only choice left is a card with three filled green diamonds in order to meet the expectations for a set, that for a set every attribute has to be either the same or different in all three cards. In the example, the card with three filled green diamonds was the only one that would have matched all of these attributes, with the same shading and shape on each card, but different numbers and colors). 
This means that we can make 81*80*1=6480 sets IF THE ORDER MATTERED. Because the order for a set does not matter, we have to divide 6480 by 6 because you can arrange three cards 6 different ways. 
6480/6=1080

For c, I got 84240 by picking cards that can't form a set. We can find this by subtracting the number of ways we can make unique sets from the number of ways to pick three cards from the deck. 
There are 81 cards in the deck, so to pick 3 we just have to do (81 choose 3) which is equal to 85320 (so there are 85320 ways to pick 3 cards from the deck). 
We know from part b that there are 1080 ways to make unique sets from the deck. 
So the answer to the problem is 85320-1080=84240

Im a little stuck on d, e, and f so a push in the right direction would be much appreciated :) 

 Jun 2, 2021
 #1
avatar+1893 
0

I love the game set. :))

(a) How many cards are in a complete deck of Set?
3^4


(b) How many unique sets are there?

81*80/3/2 = 1080

 

(c) Find the number of sets where all three cards are the same for exactly 0 attributes.
Hm, this is where we seem to disagree. 

I think you calculated the total number of 3 cards that wouldn't make a set. 

If you are unfamilar with the game, a set can be made where all the attributes are all the same or all different. 

For example, (1, red, oval hollow) (1, purple, diamond shaded) (1, green squiggle solid) would make a set because all the attributes are all different or all the same. 

 

Nice job for question a and b. :))

Your solution with b was really clever. 

 

=^._.^=

 Jun 3, 2021
 #2
avatar+34 
0

ohhhh that makes sense! would the answer be 96 for c? (4*3*2=24 for each attribute, times 4 because there are 4 attributes)

NerdyWeeb  Jun 4, 2021

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