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 India,Maurice Winternitz, History of Indian Literature, Vol. III Persia, China, Germany, and Italy.
The rows of Pascal's triangle are conventionally enumerated starting with row at the top (the 0th row). 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 (the topmost row), 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 (or any other row) is 1 (the sum of 0 and 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 " because it describes the number of combinations: the number of ways of choosing things from among a collection of things. Row numbering starts at 0, and likewise entries within a row are numbered from 0. 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 .The binomial coefficient is conventionally set to zero if k is either less than zero or greater than n. This recurrence for the binomial coefficients is known as Pascal's rule. An arbitrary binomial coefficient can be calculated as
History
The pattern of numbers that forms Pascal's triangle was known well before Pascal's time.
In India, the Chandaḥśāstra by the Ancient Indian poet and mathematician Piṅgala (3rd or 2nd century BC) describes a method of arranging two types of syllables to form metres of various lengths and counting them; as interpreted and elaborated by Pingala's 10th-century commentator Halāyudha his "method of pyramidal expansion" (meru-prastāra) for counting metres is equivalent to Pascal's triangle. Translated by S. R. Sarma from Tertiary sources: The 6th-century Indian mathematician and astronomer Varāhamihira later described a recursive method for computing binomial coefficients that is mathematically equivalent to Pascal's triangle, although arranged in a different orientation.
The Persian mathematician al-Karajī (953–1029), roughly contemporary with Halāyudha, wrote a now-lost book which contained an explicit description of Pascal's triangle. His work was preserved and repeated by the Persian polymath Omar Khayyám (1048–1131) in his Treatise on Demonstration of Problems of Algebra (c. 1070). Khayyám's distinctive contribution was to apply the triangle's binomial coefficients to a general method for extracting nth roots, an application not found in the earlier Indian or Karajian sources.. Owing to this innovation and Khayyám's broad influence on later Islamic mathematics, the triangle is known in Iran as Khayyam's triangle (). Several theorems related to the triangle were known, including the binomial theorem.
Pascal's triangle was known in China during the 11th century through the work of the Chinese mathematician Jia Xian (1010–1070). During the 13th century, Yang Hui (1238–1298) defined the triangle, and it is known as Yang Hui's triangle () in China.Weisstein, Eric W. (2003). CRC concise encyclopedia of mathematics, p. 2169. .
In Europe, Pascal's triangle appeared for the first time in the Arithmetic of Jordanus de Nemore (13th century). The binomial coefficients were calculated by Gersonides during the early 14th century, using the multiplicative formula for them.. Petrus Apianus (1495–1552) published the full triangle on the frontispiece of his book on business calculations in 1527.. Michael Stifel published a portion of the triangle (from the second to the middle column in each row) 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 (1500–1577), 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 (Treatise on Arithmetical Triangle) 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 (1708) who called it (French: Mr. Pascal's table for combinations) and Abraham de Moivre (1730) who called it (Latin: Pascal's Arithmetic Triangle), which became the basis of the modern Western name. See in particular p. 11.
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 (by mathematical induction) 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
.
(Other common notations for include , , and .) 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 (constructed by additions). 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 (taking into consideration the first row is the 0th row), 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 central 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. (The operation of repeatedly taking a convolution of something with itself is called the convolution power.)
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 (the topmost row) 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 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.. See in particular Theorem 2, which gives a generalization of this fact for all prime moduli. Every entry in row 2n − 1, n ≥ 0, is odd.. Hinz attributes this observation to an 1891 book by Édouard Lucas, Théorie des nombres (p. 420). 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 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. The 1-dimensional simplex numbers increment by 1 as the line segments extend to the next whole number along the number line. 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.
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(d) is the rising factorial.