In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after Blaise Pascal in much of the western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy.^[1] Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ... A triangle. ... Blaise Pascal (pronounced ), (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... edit See Also: Persian Empire History of Iran and Greater Iran (also referred to as the Iranian Cultural Continent by the Encyclopedia Iranica)—- consisting areas from Euphrates in the west to Indus River and Syr Darya in the east and from Caucasus, Caspian sea and Aral Sea in the north...

The rows of Pascal's triangle are conventionally enumerated starting with row zero, and the numbers in odd rows are usually staggered relative to the numbers in even rows. A simple construction of the triangle proceeds in the following manner. On the zeroth row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

This construction is related to the binomial coefficients by Pascal's rule, which states that if In mathematics, Pascals rule is a combinatorial identity about binomial coefficients. ...

is the kth binomial coefficient in the binomial expansion of (x+y)ⁿ, then In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

for any nonnegative integer n and any integer k between 0 and n.^[2]

Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices — see also pyramid, tetrahedron, and simplex. 2-dimensional renderings (ie. ... In mathematics, Pascals pyramid is a three dimensional generalization of Pascals triangle. ... In mathematics, Pascals simplex is a version of Pascals triangle of more than three dimensions. ... For other meanings, see pyramid (disambiguation). ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ...

The triangle

Here are rows zero to sixteen of Pascal's 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 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1 

Pascal's triangle and binomial expansions

Pascal's triangle determines the coefficients which arise in binomial expansions. For an example, consider the expansion In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

(x + y)² = x² + 2xy + y² = 1x²y⁰ + 2x¹y¹ + 1x⁰y².

Notice the coefficients are the numbers in row two of Pascal's triangle: 1, 2, 1. In general, when a binomial like x + y is raised to a positive integer power we have: In elementary algebra, a binomial is a polynomial with two terms: the sum of two monomials. ...

(x + y)ⁿ = a₀xⁿ + a₁xⁿ⁻¹y + a₂xⁿ⁻²y² + … + a_n−1xyⁿ⁻¹ + a_nyⁿ,

where the coefficients a_i in this expansion are precisely the numbers on row n of Pascal's triangle. In other words,

This is the binomial theorem. In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

Notice that entire right diagonal of Pascal's triangle corresponds to the coefficient of yⁿ in these binomial expansions, while the next diagonal corresponds to the coefficient of xy^n-1 and so on.

To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of (x + 1)ⁿ⁺¹ in terms of the corresponding coefficients of (x + 1)ⁿ (setting y = 1 for simplicity). Suppose then that

Now

The two summations can be reorganized as follows:

(because of how raising a polynomial to a power works, a₀ = a_n = 1)

We now have an expression for the polynomial (x + 1)ⁿ⁺¹ in terms of the coefficients of (x + 1)ⁿ (these are the a_is), which is what we need if we want to express a line in terms of the line above it. Recall that all the terms in a diagonal going from the upper-left to the lower-right correspond to the same power of x, and that the a-terms are the coefficients of the polynomial (x + 1)ⁿ, and we are determining the coefficients of (x + 1)ⁿ⁺¹. Now, for any given i not 0 or n + 1, the coefficient of the xⁱ term in the polynomial (x + 1)ⁿ⁺¹ is equal to a_i (the figure above and to the left of the figure to be determined, since it is on the same diagonal) + a_i−1 (the figure to the immediate right of the first figure). This is indeed the simple rule for constructing Pascal's triangle row-by-row.

It is not difficult to turn this argument into a proof (by mathematical induction) of the binomial theorem. In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

An interesting consequence of the binomial theorem is obtained by setting both variables x and y equal to one. In this case, we know that (1 + 1)ⁿ = 2ⁿ, and so

In other words, the sum of the entries in the nth row of Pascal's triangle is the nth power of 2.

Patterns and properties

Pascal's triangle has many properties and contains many interesting patterns of numbers.

The diagonals

Some simple patterns are immediately apparent in the diagonals of Pascal's triangle:

• The diagonals going along the left and right edges contain only 1's.
• The diagonals next to the edge diagonals contain the natural numbers in order.
• Moving inwards, the next pair of diagonals contain the triangular numbers in order.
• The next pair of diagonals contain the tetrahedral numbers in order, and the next pair give pentatope numbers. In general, each next pair of diagonals contains the next higher dimensional "d-triangle" numbers, which can be defined as

An alternative formula is as follows: In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... A triangular number is the sum of the n natural numbers from 1 to n. ... A pyramid with side length 5 contains 35 spheres. ... A pentatope number is a number in the fifth cell of any row of Pascals triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left. ...

The geometric meaning of a function tri_d is: tri_d(1) = 1 for all d. Construct a d-dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to tri_d(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find tri_d(x), have a total of x dots composing the target shape. tri_d(x) then equals the total number of dots in the shape. A 1-dimensional triangle is simply a line, and therefore tri₁(x) = x, which is the sequence of natural numbers. The number of dots in each layer corresponds to tri_d − 1(x). Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... A triangle. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

Sierpinski triangle

Image File history File links Download high-resolution version (1122x949, 3 KB) Summary Sierpinski Triangle drawn by an original recursive Java method I have written myself. ... Image File history File links Download high-resolution version (1122x949, 3 KB) Summary Sierpinski Triangle drawn by an original recursive Java method I have written myself. ...

Other patterns and properties

• The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the fractal called Sierpinski triangle, and this resemblance becomes more and more accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern is the Sierpinski triangle. More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other patterns and combinations.

• Imagine each number in the triangle is a node in a grid which is connected to the adjacent numbers above and below it. Now for any node in the grid, count the number of paths there in the grid (without backtracking) which connect this node to the top node (1) of the triangle. The answer is the Pascal number associated to that node. The interpretation of the number in Pascal's Triangle as the number of paths to that number from the tip means that on a Plinko game board shaped like a triangle, the probability of winning prizes nearer the center will be higher than winning prizes on the edges.

• The value of each row, if each number in it is considered as a decimal place and numbers larger than 9 are carried over accordingly, is a power of 11 (specifically, 11ⁿ, where n is the number of the row). For example, row two reads '1, 2, 1', which is 11² (121). In row five, '1, 5, 10, 10, 5, 1' is translated to 161051 after carrying the values over, which is 11⁵. This property is easily explained by setting x = 10 in the binomial expansion of (x + 1)^{row number}, and adjusting the values to fit in the decimal number system.

The boundary of the Mandelbrot set is a famous example of a fractal. ... Sierpinski triangle The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński who described it in 1916. ... This contestant is about to play Plinko, and can win up to $50,000. ...

More subtle patterns

There are also more surprising, subtle patterns. From a single element of the triangle, a more shallow diagonal line can be formed by continually moving one element to the right, then one element to the bottom-right, or by going in the opposite direction. An example is the line with elements 1, 6, 5, 1, which starts from the row 1, 3, 3, 1 and ends three rows down. Such a "diagonal" has a sum that is a Fibonacci number. In the case of the example, the Fibonacci number is 13: A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral. ...

 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 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1 

The second highlighted diagonal has a sum of 233. The numbers 'skipped over' between the move right and the move down-right also sum to Fibonacci numbers, being the numbers 'between' the sums formed by the first construction. For example, the numbers skipped over in the first highlighted diagonal are 3, 4 and 1, making 8.

In addition, if row m is taken to indicate row (n + 1), the sum of the squares of the elements of row m equals the middle element of row (2m − 1). For example, 1² + 4² + 6² + 4² + 1² = 70. In general form:

Another interesting pattern is that on any row m, where m is odd, the middle term minus the term two spots to the left equals a Catalan number, specifically the (m + 1)/2 Catalan number. For example: on row 5, 6 − 1 = 5, which is the 3^rd Catalan number, and (5 + 1)/2 = 3. In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. ...

Also, the sum of the elements of row m is equal to 2^m−1. For example, the sum of the elements of row 5 is 1 + 4 + 6 + 4 + 1 = 16, which is equal to 2⁴ = 16. This follows from the binomial theorem proved above, applied to (1 + 1)^m−1.

Some of the numbers in Pascal's triangle correlate to numbers in Lozanić's triangle. Lozanićs triangle (sometimes called Losanitschs triangle) is a geometric arrangement of binomial coefficients in a manner very similar to that of Pascals triangle. ...

Another interesting property of Pascal's triangle is that in rows where the second number (the 1st number following 1) is prime, all the terms in that row except the 1s are multiples of that prime.

Binomial matrix as matrix exponential (illustration for 5×5 matrices). All the dots represent 0.

Image File history File links Exp_binomial_grey_wiki. ...

The matrix exponential

See also: Pascal matrix

Due to its simple construction by factorials, a very basic representation of Pascal's triangle in terms of the matrix exponential can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, … on its subdiagonal and zero everywhere else. This page may meet Wikipedias criteria for speedy deletion. ... In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. ...

Geometric properties

Pascal's triangle can be used as a lookup table for the number of arbitrarily dimensioned elements within a single arbitrarily dimensioned version of a triangle (known as a simplex). For example, consider the 3rd line of the triangle, with values 1, 3, 3, 1. A 2-dimensional triangle has one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements (vertices, or corners). The meaning of the final number (1) is more difficult to explain (but see below). Continuing with our example, a tetrahedron has one 3-dimensional element (itself), four 2-dimensional elements (faces), six 1-dimensional elements (edges), and four 0-dimensional elements (vertices). Adding the final 1 again, these values correspond to the 4th row of the triangle (1, 4, 6, 4, 1). Line 1 corresponds to a point, and Line 2 corresponds to a line segment (dyad). This pattern continues to arbitrarily high-dimensioned hyper-tetrahedrons (simplices). In computer science, a lookup table is a data structure, usually an array or associative array, used to replace a runtime computation with a simpler lookup operation. ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... This article just presents the basic definitions. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

To understand why this pattern exists, one must first understand that the process of building an n-simplex from an (n − 1)-simplex consists of simply adding a new vertex to the latter, positioned such that this new vertex lies outside of the space of the original simplex, and connecting it to all original vertices. As an example, consider the case of building a tetrahedron from a triangle, the latter of whose elements are enumerated by row 3 of Pascal's triangle: 1 face, 3 edges, and 3 vertices (the meaning of the final 1 will be explained shortly). To build a tetrahedron from a triangle, we position a new vertex above the plane of the triangle and connect this vertex to all three vertices of the original triangle.

The number of a given dimensional element in the tetrahedron is now the sum of two numbers: first the number of that element found in the original triangle, plus the number of new elements, each of which is built upon elements of one fewer dimension from the original triangle. Thus, in the tetrahedron, the number of cells (polyhedral elements) is 0 (the original triangle possesses none) + 1 (built upon the single face of the original triangle) = 1; the number of faces is 1 (the original triangle itself) + 3 (the new faces, each built upon an edge of the original triangle) = 4; the number of edges is 3 (from the original triangle) + 3 (the new edges, each built upon a vertex of the original triangle) = 6; the number of new vertices is 3 (from the original triangle) + 1 (the new vertex that was added to create the tetrahedron from the triangle) = 4. This process of summing the number of elements of a given dimension to those of one fewer dimension to arrive at the number of the former found in the next higher simplex is equivalent to the process of summing two adjacent numbers in a row of Pascal's triangle to yield the number below. Thus, the meaning of the final number (1) in a row of Pascal's triangle becomes understood as representing the new vertex that is to be added to the simplex represented by that row to yield the next higher simplex represented by the next row. This new vertex is joined to every element in the original simplex to yield a new element of one higher dimension in the new simplex, and this is the origin of the pattern found to be identical to that seen in Pascal's triangle. A cell is a three-dimensional object that is part of a higher-dimensional object, such as a polychoron. ...

A similar pattern is observed relating to squares, as opposed to triangles. To find the pattern, one must construct an analog to Pascal's triangle, whose entries are the coefficients of (x + 2)^{Row Number}, instead of (x + 1)^{Row Number}. There are a couple ways to do this. The simpler is to begin with Row 0 = 1 and Row 1 = 1, 2. Proceed to construct the analog triangles according to the following rule: For other uses, see Square. ...

That is, choose a pair of numbers according to the rules of Pascal's triangle, but double the one on the left before adding. This results in:

 1 1 2 1 4 4 1 6 12 8 1 8 24 32 16 1 10 40 80 80 32 1 12 60 160 240 192 64 1 14 84 280 560 672 448 128 

The other way of manufacturing this triangle is to start with Pascal's triangle and multiply each entry by 2^k, where k is the position in the row of the given number. For example, the 2nd value in row 4 of Pascal's triangle is 6 (the slope of 1s corresponds to the zeroth entry in each row). To get the value that resides in the corresponding position in the analog triangle, multiply 6 by 2^{Position Number} = 6 × 2² = 6 × 4 = 24. Now that the analog triangle has been constructed, the number of elements of any dimension that compose an arbitrarily dimensioned cube (called a hypercube) can be read from the table in a way analogous to Pascal's triangle. For example, the number of 2-dimensional elements in a 2-dimensional cube (a square) is one, the number of 1-dimensional elements (sides, or lines) is 4, and the number of 0-dimensional elements (points, or vertices) is 4. This matches the 2nd row of the table (1, 4, 4). A cube has 1 cube, 6 faces, 12 edges, and 8 vertices, which corresponds to the next line of the analog triangle (1, 6, 12, 8). This pattern continues indefinitely. Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ...

To understand why this pattern exists, first recognize that the construction of an n-cube from an (n − 1)-cube is done by simply duplicating the original figure and displacing it some distance (for a regular n-cube, the edge length) orthogonal to the space of the original figure, then connecting each vertex of the new figure to its corresponding vertex of the original. This initial duplication process is the reason why, to enumerate the dimensional elements of an n-cube, one must double the first of a pair of numbers in a row of this analog of Pascal's triangle before summing to yield the number below. The initial doubling thus yields the number of "original" elements to be found in the next higher n-cube and, as before, new elements are built upon those of one fewer dimension (edges upon vertices, faces upon edges, etc.). Again, the last number of a row represents the number of new vertices to be added to generate the next higher n-cube. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

In this triangle, the sum of the elements of row m is equal to 3^m − 1. Again, to use the elements of row 5 as an example: 1 + 8 + 24 + 32 + 16 = 81, which is equal to 3⁴ = 81.

Yang Hui (Pascal's) triangle, as depicted by the Chinese using rod numerals.

Image File history File links Yanghui_triangle. ... Image File history File links Yanghui_triangle. ... Yang Hui (楊輝, c. ... The counting rods (Traditional Chinese: , Simplified Chinese: , pinyin: chou2) were used by ancient Chinese before the invention of the abacus. ...

Calculating an individual row

This algorithm is an alternative to the standard method of calculating individual cells with factorials. Starting at the left, the first cell's value is 1. For each cell after, the value is determined by multiplying the value to the left by a slowly changing fraction:

Where r = row + 1, starting with 0 at the top, and c = the column, starting with 0 on the left. For example, to calculate row 5, r=6. The first value is 1. The next value is 1 x 5/1 = 5. The numerator increases by one, and the denominator decreases by one with each step. So 5 x 4/2 = 10. Then 10 x 3/3 = 10. Then 10 x 2/4 = 5. Then 5 x 1/5 = 1. Notice that the last cell always equals 1, the final multiplication is included for completeness of the series.

A similar pattern exists on a downward diagonal. Starting with the one and the natural number in the next cell, form a fraction. To determine the next cell, increase the numerator and denominator each by one, and then multiply the previous result by the fraction. For example, the row starting with 1 and 7 form a fraction of 7/1. The next cell is 7 x 8/2 = 28. The next cell is 28 x 9/3 = 84.

History

The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an ancient Indian book on Sanskrit prosody written by Pingala between the 5th–2nd centuries BC. While Pingala's work only survives in fragments, the commentator Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru". It was also realised that the shallow diagonals of the triangle sum to the Fibonacci numbers. The Indian mathematician Bhattotpala (c. 1068) later gives rows 0-16 of the triangle. In mathematics, in particular in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here m! denotes the factorial of m). ... As a means of recording the passage of time, the 10th century was that century which lasted from 901 to 1000. ... Ancient India may refer to: The ancient History of India, which generally includes the ancient history of the whole Indian subcontinent (South Asia) Indus Valley Civilization — during the Bronze Age Vedic period — the period of Vedic Sanskrit, spanning the late Bronze Age and the earlier Iron Age Mahajanapadas — during the... Sanskrit ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... Prosody may mean several things: Prosody consists of distinctive variations of stress, tone, and timing in spoken language. ... Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... The 5th century BC started the first day of 500 BC and ended the last day of 401 BC. // The Parthenon of Athens seen from the hill of the Pnyx to the west. ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 2nd century BC started on January 1, 200 BC and ended on December 31, 101 BC. // Coin of Antiochus IV. Reverse shows Apollo seated on an omphalos. ... Halayudha (हलायुध) was a 10th century Indian mathematician, wrote a commentary on Pingalas Chandah-shastra where Pingalas knowledge of the meru-prastaara (Pascals triangle) is explicitly mentioned. ... Events Coronation of King Edward the Martyr Births Deaths July 8 Edgar of England Categories: 975 ... Mount Meru is a sacred mountain in Hindu mythology which is believed to be the abode of Brahma and other gods. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ... This article is under construction. ... Events Emperor Go-Sanjo ascends the throne of Japan William the Conqueror takes Exeter after a brief siege Births Henry I of England (d. ...

At around the same time, it was discussed in Persia (Iran) by the mathematician Al-Karaji (953–1029) and the poet-astronomer-mathematician Omar Khayyám (1048-1131); thus the triangle is referred to as the "Khayyam triangle" in Iran. Several theorems related to the triangle were known, including the binomial theorem. In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. edit See Also: Persian Empire History of Iran and Greater Iran (also referred to as the Iranian Cultural Continent by the Encyclopedia Iranica)—- consisting areas from Euphrates in the west to Indus River and Syr Darya in the east and from Caucasus, Caspian sea and Aral Sea in the north... Islamic mathematics is the profession of Muslim Mathematicians. ... Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji (953 - 1029), also known as Al-karkhi was a Persian mathematician and engineer. ... Persian literature (in Persian: ‎ ) spans two and a half millennia, though much of the pre-Islamic material has been lost. ... This is a sub-article of Islamic science and astronomy. ... For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

In 13th century, Yang Hui (1238-1298) presented the arithmetic triangle, which was the same as Pascal's Triangle. Today Pascal's triangle is called "Yang Hui's triangle" in China. Yang Hui (楊輝, c. ... The first five rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ... Yang Hui (楊輝, c. ...

Finally, in Italy, it is referred to as "Tartaglia's triangle", named for the Italian algebraist Niccolo Fontana Tartaglia who lived a century before Pascal; Tartaglia is credited with the general formula for solving cubic polynomials. Niccolo Fontana Tartaglia. ...

In 1655, Blaise Pascal wrote a Traité du triangle arithmétique (Treatise on arithmetical triangle), wherein he collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) and Abraham de Moivre (1730). Events March 25 - Saturns largest moon, Titan, is discovered by Christian Huygens. ... Blaise Pascal (pronounced ), (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... Pierre Raymond de Montmort was born in Paris on Oct. ... Abraham de Moivre. ...

References

1. ^ [1]
2. ^ The binomial coefficient is conventionally set to zero if k is either less than zero or greater than n.

See also

• bean machine, Francis Galton's "quincunx"
• Euler triangle
• Leibniz harmonic triangle
• Multiplicities of entries in Pascal's triangle (Singmaster's conjecture)
• Pascal matrix

The bean machine, also known as the quincunx or Galton box, is a device invented by Sir Francis Galton to demonstrate the law of error and the normal distribution. ... In combinatorics the Eulerian number E(n,m), or , is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m ascents). // For a given value of n, the index m in E(n,m) can take... The Leibniz harmonic triangle is a triangular arrangement of fractions in which the outermost diagonals consist of the reciprocals of the row numbers and each inner cell is the absolute value of the cell above minus the cell to the left. ... In combinatorial mathematics, it is clear that the only number that appears infinitely many times in Pascals triangle is 1. ... This page may meet Wikipedias criteria for speedy deletion. ...

External links

• Eric W. Weisstein, Pascal's triangle at MathWorld.
• The Old Method Chart of the Seven Multiplying Squares (from the Ssu Yuan Yü Chien of Chu Shi-Chieh, 1303, depicting the first nine rows of Pascal's triangle)
• Pascal's Treatise on the Arithmetic Triangle (page images of Pascal's treatise, 1655; summary: [2])
• Earliest Known Uses of Some of the Words of Mathematics (P)
• Leibniz and Pascal triangles
• Dot Patterns, Pascal's Triangle, and Lucas' Theorem
• Pascal's Triangle From Top to Bottom
• Omar Khayyam the mathematician
• Info on Pascal's Triangle
• Explanation of Pascal's Triangle and common occurrences, including link to interactive version specifying # of rows to view