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Pascal's Triangle

Tool to compute value for Pascal's triangle. Pascal's Triangle is an arithmetic list of number imagined by Blaise Pascal where each item is either 1 or the sum of the two elements above it.

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Pascal's Triangle -

Tag(s) : Arithmetics, Series

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# Pascal's Triangle

## Pascal Triangle Calculator

 Index 0 (the first row is noted 0) 1 (the first row is noted 1)

## Position of a number in Pascal Triangle

 Index 0 (the first row is noted 0)

### How to create a pascal triangle?

The principle of Pascal triangle is based on a pyramidal/triangular construction, write 1 on the first line, and 1 1 on the second line. For the next lines, take two adjacent numbers, add their values and place this new number directly above. (The missing start and end are equal to 1).

Example: Start of the Pascal Triangle:
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
1 7 21 35 35 21 7 1

Values can be calculated using binomial coefficient, also used in calculation of combinations.

Pascal triangle values can be compared to the Fibonacci sequence where each number is the sum of the two preceding numbers.

Usually, mathematicians call the first line 0, same for the first column.

### How to calculate a precise value in the Pascal's triangle?

A value $V$ of the Pascal triangle at the position (line A, column B, 0-indexed) can be calculated with the binomial coefficients (and thus with factorials) and the formula $$V = \binom{A}{B} = \frac{A!}{B!(A-B)!}$$

### How to create a Pascal Triangle with Excel?

Write 1 in the cell B1, and =A2+B1 in the cell B2 and copy the contents in as many cells as you wish but do not touch column 1 and line 1. Each line is a new row of the Pascal triangle.

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