Pascal's triangle explained

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,[1] China, Germany, and Italy.[2]

The rows of Pascal's triangle are conventionally enumerated starting with row

n=0

at the top (the 0th row). The entries in each row are numbered from the left beginning with

k=0

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

n

th row of Pascal's triangle, the

k

th entry is denoted

\tbinomnk

, pronounced " choose ". For example, the topmost entry is

\tbinom00=1

. With this notation, the construction of the previous paragraph may be written as = + for any positive integer

n

and any integer

0\lek\len

.[3] 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 (953–1029) wrote a now-lost book which contained the first formulation of the binomial coefficients and the first description of Pascal's triangle.[4] [5] [6] It was later repeated by Omar Khayyám (1048–1131), another Persian mathematician; thus the triangle is also referred to as Khayyam's triangle (Persian: مثلث خیام|label=none) in Iran.[7] 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.[8]

Pascal's triangle was known in China during the early 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 (Chinese: s=杨辉三角|t=楊輝三角|labels=no) in China.[9]

In Europe, Pascal's triangle appeared for the first time in the Arithmetic of Jordanus de Nemore (13th century).[10] The binomial coefficients were calculated by Gersonides during the early 14th century, using the multiplicative formula for them.[11] Petrus Apianus (1495–1552) published the full triangle on the frontispiece of his book on business calculations in 1527.[12] 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.[11] In Italy, Pascal's triangle is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolò Fontana Tartaglia (1500–1577), who published six rows of the triangle in 1556.[11] Gerolamo Cardano also published the triangle as well as the additive and multiplicative rules for constructing it in 1570.[11]

Pascal's French: Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published posthumously in 1665.[13] 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: table de M. Pascal pour les combinaisons (French: Mr. Pascal's table for combinations) and Abraham de Moivre (1730) who called it Latin: Triangulum Arithmeticum PASCALIANUM (Latin: Pascal's Arithmetic Triangle), which became the basis of the modern Western name.[14]

Binomial expansions

Pascal's triangle determines the coefficients which arise in binomial expansions. For example, in the expansion(x + y)^2 = x^2 + 2xy + y^2 = \mathbf x^2 y^0 + \mathbf x^1 y^1 + \mathbf x^0 y^2,the coefficients are the entries in the second row of Pascal's triangle:

\tbinom20=1

,

\tbinom21=2

,

\tbinom22=1

.

In general, the binomial theorem states that when a binomial like

x+y

is raised to a positive integer power

n

, the expression expands as(x + y)^n = \sum_^ a_ x^ y^ = a_ x^n + a_ x^ y + a_ x^ y^ + \ldots + a_ x y^ + a_ y^, where the coefficients

ak

are precisely the numbers in row

n

of Pascal's triangle:a_k = .

The entire left diagonal of Pascal's triangle corresponds to the coefficient of

xn

in these binomial expansions, while the next left diagonal corresponds to the coefficient of

xn-1y

, 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+y)n

in terms of the corresponding coefficients of

(x+1)n

, where we sett

y=1

for simplicity. Suppose then that(x + 1)^ = \sum_^ a_ x^.Now (x+1)^ = (x+1)(x+1)^n = x(x+1)^n + (x+1)^n = \sum_^n a_i x^ + \sum_^n a_k x^k.

The two summations can be reindexed with

k=i+1

and combined to yield\begin\sum_^ a_ x^ + \sum_^n a_k x^k &= \sum_^ a_ x^ + \sum_^n a_k x^k \\ [4pt] &= \sum_^ a_ x^ + a_x^ + a_0x^0 + \sum_^n a_k x^k \\[4pt] &= a_0x^0 + \sum_^ (a_ + a_k)x^ + a_x^ \\[4pt] &= x^0 + \sum_^ (a_ + a_k)x^ + x^.\end

Thus the extreme left and right coefficients remain as 1, and for any given

0<k<n+1

, the coefficient of the

xk

term in the polynomial

(x+1)n

is equal to

ak-1+ak

, the sum of the

xk-1

and

xk

coefficients in the previous power

(x+1)n

. 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

(a+b)n=bn(\tfrac{a}{b}+1)n

, the coefficients are identical in the expansion of the general case.

An interesting consequence of the binomial theorem is obtained by setting both variables

x=y=1

, so that \sum_^ = + + \cdots + + = (1+1)^n = 2^.

In other words, the sum of the entries in the

n

th row of Pascal's triangle is the

n

th power of 2. This is equivalent to the statement that the number of subsets of an

n

-element set is

2n

, as can be seen by observing that each of the

n

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

n

items taken

k

at a time, i.e. the number of subsets of

k

elements from among

n

elements, can be found by the equation

C(n,k)=

n
C
k

={nCk

} = = \frac.

This is equal to entry

k

in row of

n

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, which is

\tbinom{7}{3}=35

.[15]

Relation to binomial distribution and convolutions

When divided by

2n

, the

n

th row of Pascal's triangle becomes the binomial distribution in the symmetric case where

p=\tfrac{1}{2}

. By the central limit theorem, this distribution approaches the normal distribution as

n

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

\{\ldots,0,0,1,1,0,0,\ldots\}

with itself corresponds to taking powers of

x+1

, 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

n

equals to

2n

.

sn

for all

n\ge0

as follows:

sn=

n
\prod
k=0

{n\choosek}=

n
\prod
k=0
n!
k!(n-k)!
Then, the ratio of successive row products is \frac = \frac = \frac and the ratio of these ratios is \frac = \left(\frac \right)^n, ~ n\ge 1. The right-hand side of the above equation takes the form of the limit definition of

e

e =\lim_ \left(1 + \frac \right)^.

n=2m

, the middle term minus the term two spots to the left equals a Catalan number, specifically

Cm-1=\tbinom{2m}{m}-\tbinom{2m}{m-2}

. For example, in row 4, which is 1, 4, 6, 4, 1, we get the 3rd Catalan number

C3=6-1=5

.

\tbinompk=\tfrac{p!}{k!(p-k)!}

. Since the denominator

p!(p-k)!

can have no prime factors equal to, so remains in the numerator after integer division, making the entire entry a multiple of .

Diagonals

The diagonals of Pascal's triangle contain the figurate numbers of simplices:

\begin{align} P0(n)&=Pd(0)=1,\\ Pd(n)&=Pd(n-1)+Pd-1(n)\\ &=

n
\sum
i=0

Pd-1(i)=

d
\sum
i=0

Pi(n-1). \end{align}

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 isP_d(n)=\frac\prod_^ (n+k) = = \binom,where n(d) is the rising factorial.

The geometric meaning of a function Pd is: Pd(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 Pd(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find Pd(x), have a total of x dots composing the target shape. Pd(x) 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(x) = 1 and P1(x) = x, which is the sequence of natural numbers. The number of dots in each layer corresponds to Pd − 1(x).

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

n

with the elements

\tbinom{n}{0},\tbinom{n}{1},\ldots,\tbinom{n}{n}

, begin with

\tbinom{n}{0}=1

. For each subsequent element, the value is determined by multiplying the previous value by a fraction with slowly changing numerator and denominator:

{n\choosek}={n\choosek-1} x

n+1-k
k

.

For example, to calculate row 5, the fractions are 

\tfrac{5}{1}

\tfrac{4}{2}

\tfrac{3}{3}

\tfrac{2}{4}

and

\tfrac{1}{5}

, and hence the elements are 

\tbinom{5}{0}=1

,  

\tbinom{5}{1}=1 x \tfrac{5}{1}=5

,  

\tbinom{5}{2}=5 x \tfrac{4}{2}=10

, etc. (The remaining elements are most easily obtained by symmetry.)

To compute the diagonal containing the elements

\tbinom{n}{0},\tbinom{n+1}{1},\tbinom{n+2}{2},\ldots,

begin again with

\tbinom{n}{0}=1

and obtain subsequent elements by multiplication by certain fractions:

{n+k\choosek}={n+k-1\choosek-1} x

n+k
k

.

For example, to calculate the diagonal beginning at

\tbinom{5}{0}

, the fractions are 

\tfrac{6}{1},\tfrac{7}{2},\tfrac{8}{3},\ldots

, and the elements are

\tbinom{5}{0}=1,\tbinom{6}{1}=1 x \tfrac{6}{1}=6,\tbinom{7}{2}=6 x \tfrac{7}{2}=21

, etc. By symmetry, these elements are equal to

\tbinom{5}{5},\tbinom{6}{5},\tbinom{7}{5}

, etc.

Overall patterns and properties

As the proportion of black numbers tends to zero with increasing n, a corollary is that the proportion of odd binomial coefficients tends to zero as n tends to infinity.[20]

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

bgcolor=red1
11
1bgcolor=lime2bgcolor=aqua1
bgcolor=lime1bgcolor=aqua33bgcolor=red1
bgcolor=aqua14bgcolor=red641
1bgcolor=red51010bgcolor=lime5bgcolor=aqua1
bgcolor=red1615bgcolor=lime20bgcolor=aqua156bgcolor=red1
172135352171

Construction as 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.

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,

\R

, with basis 1.

Relation to geometry of polytopes

Pascal's triangle can be used as a lookup table for the number of elements (such as edges and corners) within a polytope (such as a triangle, a tetrahedron, a square, or a cube).

Number of elements of simplices

Let's begin by considering the 3rd line of Pascal's 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 (known as simplices).

To understand why this pattern exists, one must first understand that the process of building an n-simplex from an -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. To build a tetrahedron from a triangle, 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 ; the number of faces is the number of edges is the number of new vertices is . 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.

Number of elements of hypercubes

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:

{n\choosek}=2 x {n-1\choosek-1}+{n-1\choosek}.

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:

\begin{matrix} 1\\ 12\\ 144\\ 16128\\ 18243216\\ 11040808032\\ 1126016024019264\\ 11484280560672448128 \end{matrix}

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 (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 . 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.

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 (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 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

34=81

.

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 (1 3 3 1) corresponds to the usual three-dimensional cube: fixing a vertex V, there is one vertex at distance 0 from V (that is, V itself), three vertices at distance 1, three vertices at distance and one vertex at distance (the vertex opposite V). 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 (x + 1)n are the nth row of the triangle. Now the coefficients of (x − 1)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 sin(x)n+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 (suitably normalized) are given by the nth row of the triangle with alternating signs.[21] For example, the values of the step function that results from:

ak{Re}\left(Fourier\left[

\sin(x)5
x

\right]\right)

compose the 4th row of the triangle, with alternating signs. This is a generalization of the following basic result (often used in electrical engineering):

ak{Re}\left(Fourier\left[

\sin(x)1
x

\right]\right)

is the boxcar function.[22] 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 (normalized) first terms corresponds to the powers of i, which cycle around the intersection of the axes with the unit circle in the complex plane: +i,-1,-i,+1,+i,\ldots

Extensions

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.[23]

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

z!=\Gamma(z+1)

, Pascal's triangle can be extended beyond the integers to

\Complex

, since

\Gamma(z+1)

is meromorphic to the entire complex plane.[24]

To arbitrary bases

n

can be read as a radix

a

numeral, where

\limn

n
11
a
is the hypothetical terminal row, or limit, of the triangle, and the rows are its partial products.[26] He proved the entries of row

n

, when interpreted directly as a place-value numeral, correspond to the binomial expansion of

(a+1)n=

n
11
a
. More rigorous proofs have since been developed.[27] [28] To better understand the principle behind this interpretation, here are some things to recall about binomials:

a

numeral in positional notation (e.g.

14641a

) is a univariate polynomial in the variable

a

, where the degree of the variable of the

i

th term (starting with

i=0

) is

i

. For example,

14641a=1a4+4a3+6a2+4a1+1a0

.

(a+b)n

. The variable

b

can be eliminated from the expansion by setting

b=1

. The expansion now typifies the expanded form of a radix

a

numeral,[29] [30] as demonstrated above. Thus, when the entries of the row are concatenated and read in radix

a

they form the numerical equivalent of

(a+1)n=

n
11
a
. If

c=a+1

for

c<0

, then the theorem holds for

a=\{c-1,-(c+1)\}mod2c

with odd values of

n

yielding negative row products.[31] [32] [33]

By setting the row's radix (the variable

a

) equal to one and ten, row

n

becomes the product
n
11
1

=2n

and
n
11
10

=11n

, respectively. To illustrate, consider

a=n

, which yields the row product

nn\left(1+

1
n

\right)n=

n
11
n
. The numeric representation of
n
11
n
is formed by concatenating the entries of row

n

. The twelfth row denotes the product:
12
11
12

=1:10:56:164:353:560:650:560:353:164:56:10:112=27433a969970112

with compound digits (delimited by ":") in radix twelve. The digits from

k=n-1

through

k=1

are compound because these row entries compute to values greater than or equal to twelve. To normalize[34] the numeral, simply carry the first compound entry's prefix, that is, remove the prefix of the coefficient

{n\choosen-1}

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

01

for all

n

. The leftmost digit is

2

for

n>2

, which is obtained by carrying the

1

of

10n

at entry

k=1

. It follows that the length of the normalized value of
n
11
n
is equal to the row length,

n+1

. The integral part of
n
1.1
n
contains exactly one digit because

n

(the number of places to the left the decimal has moved) is one less than the row length. Below is the normalized value of
1234
1.1
1234
. Compound digits remain in the value because they are radix

1234

residues represented in radix ten:
1234
1.1
1234

=

1227digits:0:1
2.885:2:35:977:696:\overbrace{\ldots}
1234

=2.717181235\ldots10

See also

External links

Notes and References

  1. Maurice Winternitz, History of Indian Literature, Vol. III
  2. Book: Peter Fox . Cambridge University Library: the great collections . 1998 . Cambridge University Press . 978-0-521-62647-7 . 13.
  3. The binomial coefficient

    \scriptstyle{n\choosek}

    is conventionally set to zero if k is either less than zero or greater than n.
  4. Book: Selin, Helaine. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. 2008-03-12. Springer Science & Business Media. 9781402045592. en. 132. 2008ehst.book.....S.
  5. https://books.google.com/books?id=vSkClSvU_9AC&pg=PA62 The Development of Arabic Mathematics Between Arithmetic and Algebra - R. Rashed
  6. Book: From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren. Sidoli. Nathan. Brummelen. Glen Van. 2013-10-30. Springer Science & Business Media. 9783642367366. en. 54.
  7. Book: Kennedy, E. . Omar Khayyam. The Mathematics Teacher 1958 . i27957284. 1966 . National Council of Teachers of Mathematics . 140–142.
  8. .
  9. Weisstein, Eric W. (2003). CRC concise encyclopedia of mathematics, p. 2169. .
  10. Hughes . Barnabas . The arithmetical triangle of Jordanus de Nemore . Historia Mathematica . 1 August 1989 . 16 . 3 . 213–223 . 10.1016/0315-0860(89)90018-9 . free .
  11. .
  12. .
  13. https://gallica.bnf.fr/ark:/12148/btv1b86262012/f1.image Traité du triangle arithmétique, avec quelques autres petits traitez sur la mesme matière
  14. 10.2307/2975209 . The Binomial Coefficient Function . David . Fowler . David Fowler (mathematician) . . 103 . 1 . January 1996 . 1–17 . 2975209 . See in particular p. 11.
  15. Web site: Pascal's Triangle in Probability. 2023-06-01 . 5010.mathed.usu.edu.
  16. .
  17. .
  18. . See in particular Theorem 2, which gives a generalization of this fact for all prime moduli.
  19. . Hinz attributes this observation to an 1891 book by Édouard Lucas, Théorie des nombres (p. 420).
  20. Ian Stewart, "How to Cut a Cake", Oxford University Press, page 180
  21. For a similar example, see e.g. .
  22. .
  23. Book: Hilton, P. . etal . In International Series in Modern Applied Mathematics and Computer Science . 89–102 . Symmetry 2 . Extending the binomial coefficients to preserve symmetry and pattern . Pergamon . 1989 . 10.1016/B978-0-08-037237-2.50013-1 . 9780080372372 . https://www.sciencedirect.com/science/article/pii/B9780080372372500131 . .
  24. Book: Hilton, P. . etal . In International Series in Modern Applied Mathematics and Computer Science . 100–102 . Symmetry 2 . Extending the binomial coefficients to preserve symmetry and pattern . Pergamon . 1989 . 10.1016/B978-0-08-037237-2.50013-1 . 9780080372372 . https://www.sciencedirect.com/science/article/pii/B9780080372372500131 . .
  25. .
  26. .
  27. .
  28. .
  29. .
  30. .
  31. Book: Hilton, P. . etal . In International Series in Modern Applied Mathematics and Computer Science . 89–91 . Symmetry 2 . Extending the binomial coefficients to preserve symmetry and pattern . Pergamon . 1989 . 10.1016/B978-0-08-037237-2.50013-1 . 9780080372372 . https://www.sciencedirect.com/science/article/pii/B9780080372372500131 . .
  32. .
  33. .
  34. .