Pascal's triangle


In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, Germany, and Italy.
The rows of Pascal's triangle are conventionally enumerated starting with [|row] at the top. The entries in each row are numbered from the left beginning with and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0, there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number [|above] and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 is 1, whereas the numbers 1 and 3 in row 3 are added to produce the number 4 in row 4.

Formula

In the th row of Pascal's triangle, the th entry is denoted, pronounced " choose ". For example, the topmost entry is. With this notation, the construction of the previous paragraph may be written as
for any positive integer and any integer. This recurrence for the binomial coefficients is known as Pascal's rule.

History

The pattern of numbers that forms Pascal's triangle was known well before Pascal's time. The Persian mathematician Al-Karaji wrote a now-lost book which contained the first description of Pascal's triangle. In India, the Chandaḥśāstra by the Indian poet and mathematician Piṅgala describes a method of arranging two types of syllables to form metres of various lengths and counting them; as interpreted and elaborated by Piṅgala's 10th-century commentator Halāyudha his "method of pyramidal expansion" for counting metres is equivalent to Pascal's triangle. It was later repeated by Omar Khayyám, another Persian mathematician; thus the triangle is also referred to as Khayyam's triangle in Iran. Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.
Pascal's triangle was known in China during the 11th century through the work of the Chinese mathematician Jia Xian. During the 13th century, Yang Hui defined the triangle, and it is known as Yang Hui's triangle in China.
In Europe, Pascal's triangle appeared for the first time in the Arithmetic of Jordanus de Nemore.
The binomial coefficients were calculated by Gersonides during the early 14th century, using the multiplicative formula for them. Petrus Apianus published the full triangle on the frontispiece of his book on business calculations in 1527. Michael Stifel published a portion of the triangle in 1544, describing it as a table of figurate numbers. In Italy, Pascal's triangle is referred to as Tartaglia's triangle, named for the Italian algebraist Tartaglia, who published six rows of the triangle in 1556. Gerolamo Cardano also published the triangle as well as the additive and multiplicative rules for constructing it in 1570.
Pascal's Traité du triangle arithmétique was published posthumously in 1665. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named for Pascal by Pierre Raymond de Montmort who called it table de M. Pascal pour les combinaisons and Abraham de Moivre who called it Triangulum Arithmeticum PASCALIANUM, which became the basis of the modern Western name.

Binomial expansions

Pascal's triangle determines the coefficients which arise in binomial expansions. For example, in the expansion
the coefficients are the entries in the second row of Pascal's triangle:,,.
In general, the binomial theorem states that when a binomial like is raised to a positive integer power, the expression expands as
where the coefficients are precisely the numbers in row of Pascal's triangle:
The entire left diagonal of Pascal's triangle corresponds to the coefficient of in these binomial expansions, while the next left diagonal corresponds to the coefficient of, 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 in terms of the corresponding coefficients of, where we set for simplicity. Suppose then that
Now
The two summations can be reindexed with and combined to yield
Thus the extreme left and right coefficients remain as 1, and for any given, the coefficient of the term in the polynomial is equal to, the sum of the and coefficients in the previous power. This is indeed the downward-addition rule for constructing Pascal's triangle.
It is not difficult to turn this argument into a proof of the binomial theorem.
Since, the coefficients are identical in the expansion of the general case.
An interesting consequence of the binomial theorem is obtained by setting both variables, so that
In other words, the sum of the entries in the th row of Pascal's triangle is the th power of 2. This is equivalent to the statement that the number of subsets of an -element set is, as can be seen by observing that each of the elements may be independently included or excluded from a given subset.

Combinations

A second useful application of Pascal's triangle is in the calculation of combinations. The number of combinations of items taken at a time, i.e. the number of subsets of elements from among elements, can be found by the equation
This is equal to entry in row of Pascal's triangle. Rather than performing the multiplicative calculation, one can simply look up the appropriate entry in the triangle. For example, suppose 3 workers need to be hired from among 7 candidates; then the number of possible hiring choices is 7 choose 3, the entry 3 in row 7 of the above table, which is.

Relation to binomial distribution and convolutions

When divided by, the th row of Pascal's triangle becomes the binomial distribution in the symmetric case where. By the limit theorem">Infinite product">limit theorem, this distribution approaches the normal distribution as increases. This can also be seen by applying Stirling's formula to the factorials involved in the formula for combinations.
This is related to the operation of discrete convolution in two ways. First, polynomial multiplication corresponds exactly to discrete convolution, so that repeatedly convolving the sequence with itself corresponds to taking powers of, and hence to generating the rows of the triangle. Second, repeatedly convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for a sum of n independent copies of that variable; this is exactly the situation to which the central limit theorem applies, and hence results in the normal distribution in the limit.

Patterns and properties

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

Rows

  • The sum of the elements of a single row is twice the sum of the row preceding it. For example, row 0 has a value of 1, row 1 has a value of 2, row 2 has a value of 4, and so forth. This is because every item in a row produces two items in the next row: one left and one right. The sum of the elements of row equals to.
  • Taking the product of the elements in each row, the sequence of products is related to the base of the natural logarithm, e. Specifically, define the sequence for all as follows: Then, the ratio of successive row products is and the ratio of these ratios is The right-hand side of the above equation takes the form of the limit definition of [e (mathematical constant)|]
  • [pi|] can be found in Pascal's triangle by use of the Nilakantha infinite series.
  • Some of the numbers in Pascal's triangle correlate to numbers in Lozanić's triangle.
  • The sum of the squares of the elements of row equals the middle element of row . For example,. In general form,
  • In any even row, the middle term minus the term two spots to the left equals a Catalan number, specifically. For example, in row 4, which is 1, 4, 6, 4, 1, we get the 3rd Catalan number.
  • In a row , where is a prime number, all the terms in that row except the 1s are divisible by. This can be proven easily, from the multiplicative formula. Since the denominator can have no prime factors equal to, so remains in the numerator after integer division, making the entire entry a multiple of.
  • Parity: To count odd terms in row , convert to binary. Let be the number of 1s in the binary representation. Then the number of odd terms will be. These numbers are the values in Gould's sequence.
  • Every entry in row 2n − 1, n ≥ 0, is odd.
  • Polarity: When the elements of a row of Pascal's triangle are alternately added and subtracted together, the result is 0. For example, row 6 is 1, 6, 15, 20, 15, 6, 1, so the formula is 1 − 6 + 15 − 20 + 15 − 6 + 1 = 0.

Diagonals

The diagonals of Pascal's triangle contain the figurate numbers of simplices:
The symmetry of the triangle implies that the nth d-dimensional number is equal to the dth n-dimensional number.
An alternative formula that does not involve recursion is
where n is the rising factorial.
The geometric meaning of a function Pd is: Pd = 1 for all d. Construct a d-dimensional triangle by placing additional dots below an initial dot, corresponding to Pd = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find Pd, have a total of x dots composing the target shape. Pd then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore P0 = 1 and P1 = x, which is the sequence of natural numbers. The number of dots in each layer corresponds to Pd − 1.

Calculating a row or diagonal by itself

There are simple algorithms to compute all the elements in a row or diagonal without computing other elements or factorials.
To compute row with the elements, begin with. For each subsequent element, the value is determined by multiplying the previous value by a fraction with slowly changing numerator and denominator:
For example, to calculate row 5, the fractions are , , , and, and hence the elements are , , , etc.
To compute the diagonal containing the elements begin again with and obtain subsequent elements by multiplication by certain fractions:
For example, to calculate the diagonal beginning at, the fractions are , and the elements are, etc. By symmetry, these elements are equal to, etc.

Overall patterns and properties

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



Pascal's triangle overlaid on a grid gives the number of distinct paths to each square, assuming only rightward and downward steps to an adjacent square are considered.

  • In a triangular portion of a grid, the number of shortest grid paths from a given node to the top node of the triangle is the corresponding entry in Pascal's triangle. On a Plinko game board shaped like a triangle, this distribution should give the probabilities of winning the various prizes.
[Image:Pascal's Triangle 4 paths.svg|center|400px]
  • If the rows of Pascal's triangle are left-justified, the diagonal bands sum to the Fibonacci numbers.

Construction as matrix exponential

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 sub-diagonal and zero everywhere else.

Construction of Clifford algebra using simplices

Labelling the elements of each n-simplex matches the basis elements of Clifford algebra used as forms in Geometric Algebra rather than matrices. Recognising the geometric operations, such as rotations, allows the algebra operations to be discovered. Just as each row,, starting at 0, of Pascal's triangle corresponds to an -simplex, as described below, it also defines the number of named basis forms in dimensional Geometric algebra. The binomial theorem can be used to prove the geometric relationship provided by Pascal's triangle. This same proof could be applied to simplices except that the first column of all 1's must be ignored whereas in the algebra these correspond to the real numbers,, with basis 1.

Relation to geometry of polytopes

Each row of Pascal's triangle gives the number of elements of each dimension in a corresponding simplex. In particular, for, the th entry in the th row is the number of -dimensional elements in a -dimensional simplex. For example, a triangle one 2-dimensional element, three 1-dimensional elements, and three 0-dimensional elements ; this corresponds to the third row 1, 3, 3, 1 of Pascal's triangle. This fact can be explained by combining Pascal's rule for generating the triangle with the geometric construction of simplices: each simplex is formed from a simplex of one lower dimension by the addition of a new vertex, outside the space in which the lower-dimensional simplex lies. Then each -dimensional element in the smaller simplex remains a -dimensional element of the higher simplex, and each -dimensional element when joined to the new vertex forms a new -dimensional element of the higher simplex.
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, instead of. 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:
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:
The other way of producing this triangle is to start with Pascal's triangle and multiply each entry by 2k, 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. To get the value that resides in the corresponding position in the analog triangle, multiply 6 by. Now that the analog triangle has been constructed, the number of elements of any dimension that compose an arbitrarily dimensioned cube 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 is one, the number of 1-dimensional elements is 4, and the number of 0-dimensional elements is 4. This matches the 2nd row of the table. A cube has 1 cube, 6 faces, 12 edges, and 8 vertices, which corresponds to the next line of the analog triangle. This pattern continues indefinitely.
To understand why this pattern exists, first recognize that the construction of an n-cube from an -cube is done by simply duplicating the original figure and displacing it some distance 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. Again, the last number of a row represents the number of new vertices to be added to generate the next higher n-cube.
In this triangle, the sum of the elements of row m is equal to 3m. Again, to use the elements of row 4 as an example:, which is equal to.

Counting vertices in a cube by distance

Each row of Pascal's triangle gives the number of vertices at each distance from a fixed vertex in an n-dimensional cube. For example, in three dimensions, the third row corresponds to the usual three-dimensional cube: fixing a vertex V, there is one vertex at distance 0 from V, three vertices at distance 1, three vertices at distance and one vertex at distance . The second row corresponds to a square, while larger-numbered rows correspond to hypercubes in each dimension.

Fourier transform of sin(''x'')''n''+1/''x''

As stated previously, the coefficients of n are the nth row of the triangle. Now the coefficients of n are the same, except that the sign alternates from +1 to −1 and back again. After suitable normalization, the same pattern of numbers occurs in the Fourier transform of sinn+1/x. More precisely: if n is even, take the real part of the transform, and if n is odd, take the imaginary part. Then the result is a step function, whose values are given by the nth row of the triangle with alternating signs. For example, the values of the step function that results from:
compose the 4th row of the triangle, with alternating signs. This is a generalization of the following basic result :
is the boxcar function. The corresponding row of the triangle is row 0, which consists of just the number 1.
If n is congruent to 2 or to 3 mod 4, then the signs start with −1. In fact, the sequence of the first terms corresponds to the powers of i, which cycle around the intersection of the axes with the unit circle in the complex plane:

Extensions

Upwards

Pascal's triangle may be extended upwards, above the 1 at the apex, preserving the additive property, but there is more than one way to do so.

To higher dimensions

Pascal's triangle has higher dimensional generalizations. The three-dimensional version is known as Pascal's pyramid or Pascal's tetrahedron, while the general versions are known as Pascal's simplices.

To complex numbers

When the factorial function is defined as, Pascal's triangle can be extended beyond the integers to, since is meromorphic to the entire complex plane.

To arbitrary bases

once observed that the first five rows of Pascal's triangle, when read as the digits of an integer, are the corresponding powers of eleven. He claimed without proof that subsequent rows also generate powers of eleven. In 1964, Robert L. Morton presented the more generalized argument that each row can be read as a radix numeral, where [E (mathematical constant)|] is the hypothetical terminal row, or limit, of the triangle, and the rows are its partial products. He proved the entries of row, when interpreted directly as a place-value numeral, correspond to the binomial expansion of. More rigorous proofs have since been developed. To better understand the principle behind this interpretation, here are some things to recall about binomials:
  • A radix numeral in positional notation is a univariate polynomial in the variable, where the degree of the variable of the th term is. For example,.
  • A row corresponds to the binomial expansion of. The variable can be eliminated from the expansion by setting. The expansion now typifies the expanded form of a radix numeral, as demonstrated above. Thus, when the entries of the row are concatenated and read in radix they form the numerical equivalent of. If for, then the theorem holds for, with congruent to, and with odd values of yielding negative row products.
By setting the row's radix equal to one and ten, row becomes the product and, respectively. To illustrate, consider, which yields the row product. The numeric representation of [Superparticular ratio|] is formed by concatenating the entries of row. The twelfth row denotes the product:
with compound digits in radix twelve. The digits from through are compound because these row entries compute to values greater than or equal to twelve. To normalize the numeral, simply carry the first compound entry's prefix, that is, remove the prefix of the coefficient from its leftmost digit up to, but excluding, its rightmost digit, and use radix-twelve arithmetic to sum the removed prefix with the entry on its immediate left, then repeat this process, proceeding leftward, until the leftmost entry is reached. In this particular example, the normalized string ends with for all. The leftmost digit is for, which is obtained by carrying the of at entry. It follows that the length of the normalized value of is equal to the row length,. The integral part of contains exactly one digit because is one less than the row length. Below is the normalized value of. Compound digits remain in the value because they are radix residues represented in radix ten: