Deck in the game Set

Question

In the card game Set, each card features a number of shapes, with four attributes: There is only 1 card for each possible combination of attributes. For more information about the game: go here.

Questions:

1. Find the number of sets where all three cards are the same for exactly $0$ attributes.

2. Find the number of sets where all three cards are the same for exactly $1$ attribute.

3. Find the number of sets where all three cards are the same for exactly $2$ attribute.

4. Find the number of sets where all three cards are the same for exactly $3$ attribute.

Answer 1

Hint: Each possible set with $k$ attributes that are the same can be uniquely constructed as follows:

1. Select $k$ out of $4$ attributes that will be the same,
2. For each of these, select the common "value" (e.g. if shape is a common attribute, select the shape),
3. For each remaining attribute, assign each of the three available options to one of the cards.

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For example, in the case where $k = 3$, each set can be constructed with the following procedure

1. We first select $3$ out of $4$ attributes that will be the same; there are $\binom 4k = \binom 43 = 4$ ways to do so
2. We then select one of $3$ possibilities for each of the $k = 3$ attributes. There are $3^k = 3^3 = 27$ ways to do so.
3. There is $4 - k = 1$ remaining attribute. We distribute the $3$ possible values for this attribute among the $3$ indistinguishible cards. There are $(3!)^{4 - k - 1} =1$ way of doing so.

Thus, we find that all together there are $$ 4 \times 27 = 108 $$ sets that can be constructed in which $3$ attributes are the same.