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Pascal's Triangle

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.

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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)

Calculate a Binomial Coefficient

What is Pascal triangle? (Definition)

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.

How to create a pascal triangle?

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.

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

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 $$V = \binom{A}{B} = \frac{A!}{B!(A-B)!}$$

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

Why is the triangle named Pascal?

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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