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Permutations and Combinations Compared

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Permutations and Combinations Compared - Answer Key

Using Permutations and Combinations to Compare Sudoku and Big Two

Topic

Using Permutations and Combinations to Compare Sudoku and Big Two

Vocabulary

  • Permutations
  • Combinations
  • Permutations with Repetition

Student Exploration

Have you ever wondered how many different games are possible in a Sudoku puzzle? How many different ways can you be dealt a hand of cards in Big Two? And which game is a Permutation and which is a Combination?

A Sudoku game.

Sudoku is a logic-based, number-placement puzzle. The board is a 9-by-9 square and is partially completed. The goal of the game is to place the numbers 1 to 9 in each row, each column, and each of the 3-by-3 sub-boxes. For more information on the game visit, http://en.wikipedia.org/wiki/Sudoku.

How many different possible ways are there to play in the first row of numbers?
You can use the counting principle to determine the number of possible ways to make a play in the first row: 9!= 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=362880. You could also use Permutation notation as follows: _9P_9 = \frac{9!}{(9-9!)}=\frac{9!}{0!}=9!= 362880.
How many different possible ways are there to play in first and second row of numbers?
There are still 362880 possibilities for the first row but in the second row you can’t repeat any of the same numbers in the same columns as the numbers in the first row. Therefore there are following possibilities for the second row: 8!= 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot 1=40320. Then, to find the different possibilities for the first and second row you multiply the possibilities for each row, 362880 by 40320, which equals 14,631,321,600.
How many different possible ways are there to play the entire game of Sudoku (all nine rows)?
To get the total number of possible ways to play in the entire game of Sudoku (all nine rows) multiplies the possibilities in each row. 1^{st} row: 362880, 2^{nd} row: 40320, 3^{rd} row: 5040, 4^{th} row: 720, 5^{th} row: 120, 6^{th} row: 24, 7^{th} row: 6, 8^{th} row: 2, and 9^{th} row: 1. There are a grand total of 2,548,518,711,000,000,000 ways to play the entire game of Sudoku. There are a lot of ways!

Playing Cards.

The game Big Two, also called Deuces, is a game where all the cards in a deck of 52 cards are dealt to four players. As a result each player gets 13 cards. Then the player with the three of diamonds plays first and rotates to the rest of the players in a counter clockwise direction. Players must play a card or group of cards that are ranked higher than the previous card/s played. The ranking of the cards is specific with the twos holding the highest value and the suits of cards having different values, see the links below for more information. The goal is to be the first player to get rid of all your cards, causing the game to end, and the rest of the players want to have as few cards as possible at the end of the game. For more information on the game visit, http://en.wikipedia.org/wiki/Big_Two or http://www.pagat.com/climbing/bigtwo.html.

How many different ways can you be dealt a hand of cards in the game of Big Two?
The number of ways you can be dealt a hand of cards in Big Two is calculated with _{52}C_{13} = \frac{52!}{(52-13)! 13!}=\frac{52!}{39! 13!}=\frac{52 \cdot 51 \cdot 50 \cdot \ldots \cdot 41 \cdot 40}{13!}=524,323,122,600

The game of Big Two starts with the three of diamonds being played.

Extension Investigation

  1. Why dos the Sudoku problem represent a Permutation problem, rather than a Combination? Explain. The Sudoku problem is a Permutation problem because the order that you put the numbers in one row makes a difference. If you start the row with 1, 5, 9, ... that is a different play than 9, 1, 5, ... The objective of the game is to find the correct placement of the numbers, if you could put them in any order it would be a really easy game.
  2. Why does the number of ways the cards can be dealt in the game of Big Two represent a Combination problem, rather than a Permutation? Explain. The ways of being dealt the cars in Big Two is a Combination problem because the order that you are dealt the cards doesn’t make a difference. Being dealt a hand with a two of hearts and then a Queen of spades is the same hand as a Queen of spades and a two of hearts.
  3. Derive the formula for Combinations. Show your work.
  4. What would you do differently to derive the formula for Permutations? Explain.

Resources Cited

http://en.wikipedia.org/wiki/Sudoku

http://en.wikipedia.org/wiki/Big_Two

http://www.pagat.com/climbing/bigtwo.html

Connections to other CK-12 Subject Areas

  • Probability and Permutations
  • Permutation Problems
  • Probability and Combinations
  • Combinations
  • Combination Problems.

Image from:

http://www.shutterstock.com/language.en/pic.mhtml?id=40396777&rid=945688

http://pixabay.com/en/game-six-cards-play-hearts-34749/

http://pixabay.com/en/card-recreation-games-cards-28337/

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