# Difference between revisions of "Pascal's triangle"

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− | == | + | <div class="thumb tright"> |

− | Pascal's triangle is a triangle in | + | <div> |

+ | <div> | ||

+ | <table border="0"> | ||

+ | <tr><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td></td><td></td><td>1</td><td></td><td>2</td><td></td><td>1</td><td></td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td></td><td>1</td><td></td><td>3</td><td></td><td>3</td><td></td><td>1</td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td>1</td><td></td><td>4</td><td></td><td>6</td><td></td><td>4</td><td></td><td>1</td><td></td><td></td></tr> | ||

+ | <tr><td></td><td>1</td><td></td><td>5</td><td></td><td>10</td><td></td><td>10</td><td></td><td>5</td><td></td><td>1</td><td></td></tr> | ||

+ | <tr><td>1</td><td></td><td>6</td><td></td><td>15</td><td></td><td>20</td><td></td><td>15</td><td></td><td>6</td><td></td><td>1</td></tr> | ||

+ | </table> | ||

+ | </div> | ||

+ | <div class="thumbcaption">First few rows of Pascal's Triangle</div> | ||

+ | </div> | ||

+ | </div> | ||

+ | |||

+ | '''Pascal's triangle''' is a triangle which contains the values from the [[binomial expansion]]; its various properties play a large role in [[combinatorics]]. | ||

+ | |||

+ | == Properties == | ||

+ | |||

+ | === Binomial coefficients === | ||

+ | Pascal's Triangle is defined such that the number in row <math>n</math> and column <math>k</math> is <math>\displaystyle{n\choose k}</math>. For this reason, convention holds that both row numbers and column numbers start with 0. Thus, the apex of the triangle is row 0, and the first number in each row is column 0. As an example, the number in row 4, column 2 is <math>\displaystyle{4 \choose 2} = 6</math>. Pascal's triangle thus can serve as a "look-up table" for binomial expansion values. Also, many of the characteristics of Pascal's triangle are derived from identities involving binomial expansions; for example, because <math>\sum_{k=0}^{n}{{n \choose k}}=2^n</math>, the sum of the values on row <math>n</math> of Pascal's triangle is <math>2^n</math>. | ||

+ | |||

+ | === Sum of previous values === | ||

+ | |||

+ | <div class="thumb tright"> | ||

+ | <div> | ||

+ | <div> | ||

+ | <table border="0"> | ||

+ | <tr><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td></td><td></td><td>1</td><td></td><td>2</td><td></td><td>1</td><td></td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td></td><td>1</td><td></td><td>3</td><td></td><td>3</td><td></td><td>1</td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td>1</td><td></td><td>4</td><td></td><td>6</td><td></td><td>4</td><td></td><td>1</td><td></td><td></td></tr> | ||

+ | <tr><td></td><td>1</td><td></td><td>'''5'''</td><td></td><td>'''10'''</td><td></td><td>10</td><td></td><td>5</td><td></td><td>1</td><td></td></tr> | ||

+ | <tr><td>1</td><td></td><td>6</td><td></td><td>'''15'''</td><td></td><td>20</td><td></td><td>15</td><td></td><td>6</td><td></td><td>1</td></tr> | ||

+ | </table> | ||

+ | </div> | ||

+ | <div class="thumbcaption">Sum of previous values</div> | ||

+ | </div> | ||

+ | </div> | ||

+ | One of the best known features of Pascal's triangle is derived from the combinatorics identity <math>{n \choose k}+{n \choose k+1} = {n+1 \choose k+1}</math>. Thus, any number in the interior of Pascal's triangle will be the sum of the two numbers appearing above it. For example, <math>{5 \choose 1}+{5 \choose 2} = 5 + 10 = 15 = {6 \choose 2}</math>, as shown in the diagram. This property allows the easy creation of the first few rows of Pascal's triangle without having to calculate out each binomial expansion. | ||

− | == | + | === Fibonacci numbers === |

− | |||

<div class="thumb tright"> | <div class="thumb tright"> | ||

− | <div | + | <div> |

− | + | <div> | |

− | + | <table border="0"> | |

− | 1 | + | <tr><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td></tr> |

− | + | <tr><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr> | |

− | + | <tr><td></td><td></td><td></td><td></td><td>1</td><td></td><td>2</td><td></td><td>1</td><td></td><td></td><td></td><td></td></tr> | |

− | + | <tr><td></td><td></td><td></td><td>1</td><td></td><td>3</td><td></td><td>3</td><td></td><td>'''1'''</td><td></td><td></td><td></td></tr> | |

− | <div class="thumbcaption"> | + | <tr><td></td><td></td><td>1</td><td></td><td>4</td><td></td><td>'''6'''</td><td></td><td>4</td><td></td><td>1</td><td></td><td></td></tr> |

− | + | <tr><td></td><td>1</td><td></td><td>'''5'''</td><td></td><td>10</td><td></td><td>10</td><td></td><td>5</td><td></td><td>1</td><td></td></tr> | |

+ | <tr><td>'''1'''</td><td></td><td>6</td><td></td><td>15</td><td></td><td>20</td><td></td><td>15</td><td></td><td>6</td><td></td><td>1</td></tr> | ||

+ | </table> | ||

+ | </div> | ||

+ | <div class="thumbcaption">Shallow diagonals</div> | ||

+ | </div> | ||

+ | </div> | ||

+ | |||

+ | The [[Fibonacci numbers]] appear in Pascal's Triangle along the [[shallow diagonals]]; explicitly; <math>{n \choose 0}+{n-1 \choose 1}+\cdots+{n-\lfloor\frac{n}{2}\rfloor \choose \lfloor \frac{n}{2} \rfloor} = F(n+1)</math>, where <math>F(n)</math> is the Fibonacci sequence. For example, <math>{6 \choose 0}+{5 \choose 1}+{4 \choose 2}+{3 \choose 3} = 1 + 5 + 6 + 2 = 13 = F(7)</math>. The "shallow diagonal" for this expression is plotted in the diagram. | ||

+ | |||

+ | |||

+ | === Hockey-stick theorem === | ||

+ | |||

+ | <div class="thumb tright"> | ||

+ | <div> | ||

+ | <div> | ||

+ | <table border="0"> | ||

+ | |||

+ | |||

+ | <tr><td></td><td></td><td></td><td></td><td>'''1'''</td><td></td><td>2</td><td></td><td>1</td><td></td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td></td><td>1</td><td></td><td>'''3'''</td><td></td><td>3</td><td></td><td>1</td><td></td><td></td><td></td></tr> | ||

+ | <tr><td></td><td></td><td>1</td><td></td><td>4</td><td></td><td>'''6'''</td><td></td><td>4</td><td></td><td>1</td><td></td><td></td></tr> | ||

+ | <tr><td></td><td>1</td><td></td><td>5</td><td></td><td>10</td><td></td><td>'''10'''</td><td></td><td>5</td><td></td><td>1</td><td></td></tr> | ||

+ | <tr><td>1</td><td></td><td>6</td><td></td><td>15</td><td></td><td>'''20'''</td><td></td><td>15</td><td></td><td>6</td><td></td><td>1</td></tr> | ||

+ | </table> | ||

+ | </div> | ||

+ | <div class="thumbcaption">Hockey-stick Theorem</div> | ||

</div> | </div> | ||

</div> | </div> | ||

− | + | The [[Hockey-stick theorem]] states: | |

− | + | <math>{n \choose 0}+{n+1 \choose 1}+\cdots+{n+k \choose k} = {n+k+1 \choose k}</math>. Its name is due to the "hockey-stick" which appears when the numbers are plotted on Pascal's Triangle, as shown in the representation of the theorem to the right (where <math>n=2</math> and <math>k=3</math> | |

− | === | + | === Number Parity === |

− | + | Consider writing the row number <math>n</math> in base two as <math>({n})_{10} = {(a_xa_{x-1} \cdots a_1a_0)}_2</math><math> = a_x 2^x+a_{x-1} 2^{x-1}+\cdots+a_1 2^1+a_0 2^0</math>. The number in the <math>k</math>th column of the <math>n</math>th row in Pascal's triangle is odd [[if and only if]] <math>k</math> can be expressed as the sum of some <math>a_i 2^i</math>. For example, <math>(9)_{10} = {(1001)}_{2} = 2^{3}+2^{0}</math>. Thus, the only 4 odd numbers in the 9th row will be in the <math>{(0000)}_{2} = 0</math>th, <math>{(0001)}_{2} = 2^0 = 1</math>st, <math>{(1000)}_{2} = 2^3 = 8</math>th, and <math>{(1001)}_{2} = 2^3+2^0 = 9</math>th columns. Additionally, marking each of these odd numbers in Pascal's triangle creates a [[Sierpinski triangle]]. | |

==See Also== | ==See Also== | ||

*[[Binomial theorem]] | *[[Binomial theorem]] |

## Revision as of 22:29, 24 June 2006

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1 | 6 | 15 | 20 | 15 | 6 | 1 |

**Pascal's triangle** is a triangle which contains the values from the binomial expansion; its various properties play a large role in combinatorics.

## Contents

## Properties

### Binomial coefficients

Pascal's Triangle is defined such that the number in row and column is . For this reason, convention holds that both row numbers and column numbers start with 0. Thus, the apex of the triangle is row 0, and the first number in each row is column 0. As an example, the number in row 4, column 2 is . Pascal's triangle thus can serve as a "look-up table" for binomial expansion values. Also, many of the characteristics of Pascal's triangle are derived from identities involving binomial expansions; for example, because , the sum of the values on row of Pascal's triangle is .

### Sum of previous values

1 | ||||||||||||

1 | 1 | |||||||||||

1 | 2 | 1 | ||||||||||

1 | 3 | 3 | 1 | |||||||||

1 | 4 | 6 | 4 | 1 | ||||||||

1 | 5 | 10 | 10 | 5 | 1 | |||||||

1 | 6 | 15 | 20 | 15 | 6 | 1 |

One of the best known features of Pascal's triangle is derived from the combinatorics identity . Thus, any number in the interior of Pascal's triangle will be the sum of the two numbers appearing above it. For example, , as shown in the diagram. This property allows the easy creation of the first few rows of Pascal's triangle without having to calculate out each binomial expansion.

### Fibonacci numbers

1 | ||||||||||||

1 | 1 | |||||||||||

1 | 2 | 1 | ||||||||||

1 | 3 | 3 | 1 | |||||||||

1 | 4 | 6 | 4 | 1 | ||||||||

1 | 5 | 10 | 10 | 5 | 1 | |||||||

1 | 6 | 15 | 20 | 15 | 6 | 1 |

The Fibonacci numbers appear in Pascal's Triangle along the shallow diagonals; explicitly; , where is the Fibonacci sequence. For example, . The "shallow diagonal" for this expression is plotted in the diagram.

### Hockey-stick theorem

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1 | 1 | |||||||||||

1 | 2 | 1 | ||||||||||

1 | 3 | 3 | 1 | |||||||||

1 | 4 | 6 | 4 | 1 | ||||||||

1 | 5 | 10 | 10 | 5 | 1 | |||||||

1 | 6 | 15 | 20 | 15 | 6 | 1 |

The Hockey-stick theorem states: . Its name is due to the "hockey-stick" which appears when the numbers are plotted on Pascal's Triangle, as shown in the representation of the theorem to the right (where and

### Number Parity

Consider writing the row number in base two as . The number in the th column of the th row in Pascal's triangle is odd if and only if can be expressed as the sum of some . For example, . Thus, the only 4 odd numbers in the 9th row will be in the th, st, th, and th columns. Additionally, marking each of these odd numbers in Pascal's triangle creates a Sierpinski triangle.