Graph theory. Please write neatly The last two problems require Hall\'s Theorem

Question

Graph theory. Please write neatly

The last two problems require Hall's Theorem (3.1.11), which we only discussed briefly at the end of class on Thursday 2/22/2018. We will discuss Hall's Theorem in detail on Tuesday 02/27/2018. 4: A standard card deck has 52 cards with 4 suits (hearts, spades, diamonds and clubs) and 13 values. The o an array with 4 rows and 13 columms. Prove that there is a set S of 13 cards such that S contains exactly one card from each column and every one of the 13 possible values appears (and is not repeated) in the set S.

Explanation / Answer

You have 52 cards, divided like this

Now, you have the set S contains 13 cards, so we have 13 spaces for different cards, and you can count the number of possibilities in each space

____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____  

4 8 12 16 20 24 28 32 36 40 44 48 52

Now, you can choose any of the cards, but knowing that you cannot repeat them and the 13 values need to be present. We can divide it like this.

Hearts 1 2 3 4 5 6 7 8 9 J Q K A Spades 1 2 3 4 5 6 7 8 9 J Q K A Diamonds 1 2 3 4 5 6 7 8 9 J Q K A Clubs 1 2 3 4 5 6 7 8 9 J Q K A

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