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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. When three cards form a set, we can also count the number of attributes for which all three cards are the same.



(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.

Answer :

A) The number of cards that are in a complete deck of Set is; 81

B) The number of unique sets are; 36

How to find the combination of a set of cards?

a) We are given;

Number of shapes = 1, 2 or 3

Colour = Red, Purple or Green

Shape = Oval, Diamond or Squiggle

Shading = Hollow, Shaded or Striped

Now, since a set is formed by all 3 cards, then we can say that;

Number of cards in a complete deck of set = 3⁴ = 81 cards

(b) Let us first find all sets = 3² * 4 * 3 = 108.

Thus;

Unique sets = 108/3

= 36 unique sets.

(c) There are 3 ways to choose the number of the cards from the given order.  

For the second one, there 2 choices, and then there is only 1 choice.  Thus; Number of ways to choose the numbers = 3 * 2 * 1 = 6.  

For the other attributes;

Number of ways = 6 * 6 * 24 *6 ways.

Now, since the order of cards doesn't matter in a set, then the number of sets where all three cards are the same for exactly 0 attributes is;

(6 * 6 * 24 *6)/3! = 864 sets

(d) If all the colors are the same, then;

Number of sets = 6 * 24 * 6/3! = 144 sets.  

If all the numbers are the same, then there are 144 sets which is also same for shapes and shading. Thus;

The number of sets where all three cards are the same for exactly 1 attribute is; 4 * 144 = 576

(e) There are C(4, 2) = 6 ways of choosing two attributes.  

For each of the two attributes, there are 3 options.  

For the other two attributes, there are 3 ways of assigning the choices. Thus, number of sets = 6*3*3*3*3 = 486 sets.

(f) There are C(4,3) = 4 ways of choosing which attributes are the same.  There are 3*3*4 = 36 ways to assign which is which for each of these three attributes, and there are 4 ways to assign the choices for the fourth attribute. Thus;

The number of sets where all three cards are the same for exactly 3 attributes = 4*36*4 = 576 sets.

Read more about Card combinations at; https://brainly.com/question/27447394

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