Tool to compute value for Pascal's triangle, an arithmetic list of numbers where each item is either 1 or the sum of the two elements above it.

Pascal's Triangle - dCode

Tag(s) : Arithmetics, Series

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Pascal's triangle is a representation in a triangular grid in which each number is the sum of the 2 numbers above it. In more mathematical terms, Pascal's triangle represents the binomial coefficients.

The principle of the Pascal triangle is based on a pyramidal/triangular construction, write 1 on the first row, and 1 1 on the second row. For the next rows, 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 coefficients, 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 row `0`, same for the first column.

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

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 row 1. Each row (including zeros) is a new row of the Pascal triangle.

The triangle is named in honor of Blaise Pascal, who studied it. Although he was not the first to study it, his name is the most used, although it is also called the Khayyam or Tartaglia triangle.

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Cite as source (bibliography):

*Pascal's Triangle* on dCode.fr [online website], retrieved on 2024-11-05,

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pascal,triangle,binomial,combination,sum,khayyam

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