SOLUTION: Problem: In the game of SET,Find the number of sets where all three cards are the same for exactly 0,1,2,3 attributes. Background info(in case you don't know what the game se

Question 1196898: Problem:
In the game of SET,Find the number of sets where all three cards are the same for exactly 0,1,2,3 attributes.

Background info(in case you don't know what the game set is):
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 striped.
There is exactly one card for each possible combination of attributes.
In the game, several of the cards are dealt out, and the goal is to find a set. A set is formed by three cards, where for each attribute, either all three cards are the same, or all three cards are different.

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
To solve this problem, we can break it down into cases based on the number of attributes that are the same for all three cards in a set.
**Case 1: 0 matching attributes**
In this case, all four attributes must be different for each card. For the first card, we have 3 choices for each attribute. For the second card, we have 2 choices for each attribute (since it can't be the same as the first card). For the third card, we have 1 choice for each attribute (it must be different from the first two).
So, the number of sets with 0 matching attributes is:
3 * 3 * 3 * 3 * 2 * 2 * 2 * 2 * 1 * 1 * 1 * 1 = 3^4 * 2^4 = 54648
**Case 2: 1 matching attribute**
We can choose one of the four attributes to be the same for all three cards. For the chosen attribute, we have 3 choices. For the other three attributes, we can use the same logic as in Case 1.
So, the number of sets with 1 matching attribute is:
4 * 3 * 3 * 3 * 2 * 2 * 2 * 2 * 1 * 1 * 1 * 1 = 4 * 3^3 * 2^4 = 26352
**Case 3: 2 matching attributes**
We can choose two of the four attributes to be the same for all three cards. For each of the chosen attributes, we have 3 choices. For the other two attributes, we can use the same logic as in Case 1.
So, the number of sets with 2 matching attributes is:
(4 choose 2) * 3 * 3 * 2 * 2 * 1 * 1 * 1 * 1 = 6 * 3^2 * 2^2 = 4212
**Case 4: 3 matching attributes**
We can choose three of the four attributes to be the same for all three cards. For each of the chosen attributes, we have 3 choices. For the remaining attribute, we have 3 choices.
So, the number of sets with 3 matching attributes is:
(4 choose 3) * 3 * 3 * 3 = 4 * 3^3 = 108
Therefore, the total number of sets with 0, 1, 2, or 3 matching attributes is:
54648 + 26352 + 4212 + 108 = 85320

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