Binomial coefficients, denoted by
are so named because
they appear as coefficient in the expansion of
. In particular
is the coefficient of the terms
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 )
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