## 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?
- How many different possible ways are there to play in first and second row of numbers?
- How many different possible ways are there to play the entire game of Sudoku (all nine rows)?

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

### Extension Investigation

- Why dos the Sudoku problem represent a Permutation problem, rather than a Combination? Explain.
- 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.
- Derive the formula for Combinations. Show your work.
- 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:

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