Binomial coefficients, denoted by
or
are so named because
they appear as coefficient in the expansion of
. In particular
is the coefficient of the terms
and
.
It holds
Binomial coeffients arise in many mathematical problems, especially combinatoris. For example, there are ways to choose elements from a set of objects.
The first nontrivial binomial coefficients (those for which )
are
6, 10, 15, 20, 21, 28, 35, 36, 45, 55, 56, 66, 70, 78, 84, 91, 105,
120, 126, 136, 153, 165, 171, 190, 210, 220 more terms
Below, the spiral pattern of nontrivial binomial coefficients up to . See the page on prime numbers for an explanation and links to similar pictures.
The airy spirals that appear in the drawing are caused by the
triangular numbers, which are a subset of binomial coefficients.
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details
click here
A graph displaying how many binomial coefficients are multiples of the primes
p from 2 to 71. In black the ideal line 1/
p.