binomial coefficient

binomial coefficient

For

integers

n

≥

r

≥

0

we define

(

n

r

)

=

n

!

(

n

-

r

)

!

⁢

r

!

and call such numbers

binomial coefficients

.

Properties.

1.

(

n

r

)

is an integer (proof. (

http://planetmath.org/NchooseRIsAnInteger

)).

2.

(

n

r

)

=

(

n

n

-

r

)

.

3.

(

n

r

-

1

)

+

(

n

r

)

=

(

n

+

1

r

)

(Pascal’s rule).

4.

(

n

0

)

=

1

=

(

n

n

)

for all

n

.

5.

(

n

0

)

+

(

n

1

)

+

(

n

2

)

+

⋯

+

(

n

n

)

=

2

n

.

6.

(

n

0

)

-

(

n

1

)

+

(

n

2

)

-

⋯

+

(

-

1

)

n

⁢

(

n

n

)

=

0

for

n

>

0

.

7.

∑

t

=

k

n

(

t

k

)

=

(

n

+

1

k

+

1

)

.

Properties

5

and

6

are the

binomial theorem

applied to

(

1

+

1

)

n

and

(

1

-

1

)

n

, respectively, although they also have purely combinatorial meaning.

Motivation

Suppose

n

≥

r

are integers.
The below list shows some examples where the binomial coefficients
appear.

•

(

n

r

)

constitute the coefficients when expanding the

binomial

(

x

+

y

)

n

– hence the name binomial coefficients.
See Binomial Theorem.

•

(

n

r

)

is the number of ways to choose

r

objects from a set with

n

elements

.

•

On the

context

of computer science, it also helps to see

(

n

r

)

as the number
of strings consisting of ones and zeros with

r

ones and

n

-

r

zeros.
This equivalency comes from the fact that
if

S

be a

finite set

with

n

elements,

(

n

r

)

is the number of distinct
subsets of

S

with

r

elements. For each subset

T

of

S

, consider the

function

X

T

:

S

→

{

0

,

1

}

where

X

T

⁢

(

x

)

=

1

whenever

x

∈

T

and

0

otherwise (so

X

T

is the

characteristic function

for

T

). For each

T

∈

𝒫

⁢

(

S

)

,

X

T

can be used to produce a unique bit
string of length

n

with exactly

r

ones.

Notes

The

(

n

r

)

notation was first introduced by
von Ettinghausen

[

1

]

in 1826, altough
these numbers have been used long before that. See
this page (

http://planetmath.org/PascalsTriangle

) for some notes on their history.
Although the standard mathematical notation for the binomial coefficients is

(

n

r

)

, there
are also several variants. Especially in high school environments one encounters also

C

⁢

(

n

,

r

)

or

C

r

n

for

(

n

r

)

.

Remark

. It is sometimes convenient to set

(

n

r

)

:=

0

when

r

>

n

. For example, property 7 above can be restated:

∑

t

=

1

n

(

t

k

)

=

(

n

+

1

k

+

1

)

. It can be shown that

(

n

r

)

is

elementary recursive

.

References

1

N. Higham,

Handbook of writing for the mathematical sciences

,
Society for Industrial and Applied Mathematics, 1998.

Title

binomial coefficient

Canonical name

BinomialCoefficient

Date of creation

2013-03-22 11:47:25

Last modified on

2013-03-22 11:47:25

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

32

Author

matte (1858)

Entry type

Definition

Classification

msc 11B65

Classification

msc 05A10

Classification

msc 19D55

Classification

msc 19K33

Classification

msc 19D10

Synonym

combinations

Synonym

choose

Related topic

PascalsRule

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MultinomialTheorem

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ProofOfLucassTheorem2

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Factorial

Related topic

CentralBinomialCoefficient

Related topic

PascalsTriangle

Related topic

TaylorSeriesViaDivision

Related topic

CombinationsWithRepeatedElements

Related topic

NonIsomorphicGroupsOfGivenOrder

Related topic

AppellSeque

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