The binomial coefficient of the natural number n and the integer k is defined to be the natural number
For example,
The important recurrence relation
row 0 1 row 1 1 1 row 2 1 2 1 row 3 1 3 3 1 row 4 1 4 6 4 1 row 5 1 5 10 10 5 1 row 6 1 6 15 20 15 6 1 row 7 1 7 21 35 35 21 7 1 row 8 1 8 28 56 70 56 28 8 1Row number n contains the numbers C(n, k) for k = 0,...,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that
The triangle was described by Zhu Shijie in 1303 AD in his book Precious Mirror of the Four Elements. In his book, Zhu mentioned the triangle as an ancient method (over 200 years before his time) for solving binomial coefficients, which indicated that the method was known to Chinese mathematicians five centuries before Pascal.
If you color in all even numbers on this triangle and leave the odd numbers blank, you get the Sierpinski triangle. Try coloring in multiples of 3, 4, 5, and so on and see what patterns emerge!
Table of contents |
2 Formulas involving binomial coefficients 3 Divisors of binomial coefficients 4 Generalization to complex arguments |
Binomial coefficients are of importance in combinatorics, because they provide ready formulas for certain frequent counting problems:
Combinatorics and statistics
The binomial coefficients also occur in the formula for the binomial distribution in statistics and in the formula for a Bézier curve.
The following formulas are occasionally useful:
Formulas involving binomial coefficients
C(n, k) = C(n, n-k) (4)
This follows from expansion (2) by using (x + y)n = (y + x)n.
From expansion (2) using x = y = 1.
From expansion (2), after differentiating and substituting x = y = 1.
The prime divisors of C(n, k) can be interpreted as follows: if p is a prime number and pr is the highest power of p which divides C(n, k), then r is equal to the number of natural numbers j such that the fractional part of k/pj is bigger than the fractional part of n/pj. In particular, C(n, k) is always divisible by n/gcd(n,k).
The binomial coefficient C(z, k) can be defined for any complex number z and any natural number k as follows:
For fixed k, the expression C(z, k) is a polynomial in z of degree k with rational coefficients. Every polyomial p(z) of degree d can be written in the form
Divisors of binomial coefficients
Generalization to complex arguments
This generalization is used in the formulation of the binomial theorem and satisfies properties (3) and (7).
with suitable constants ak. This is important in the theory of difference equations and can be seen as a discrete analog of Taylor's theorem.