## Accurate Binomial Coefficients

Version 1.0.2 (5.54 MB) by
Binomial coefficients / Pascal Triangle elements for / N choose K look-up table (LUT) with no numerical errors for 0 <= N <=1029.

Updated 15 Apr 2022

Syntax
b = binomial(n)
b = binomial(n, k)
b = binomial(n, [])
Description
b = binomial(n) generates a table with the binomial coefficients up to order 'n', with 0 <= n <= 1029.
b = binomial(n, k) returns the 'N' over 'K' coefficients in n(:) and k(:) or in n(:, 1), n(:, 2).
b = binomial(n, []) returns the 'N' + 1 -th row of the Pascal Triangle, or the rows in n(:) + 1.
The values are pre-computed using arbitrary numerical precision and therefore are exempt of overflows. Due to the numerical precision of the 'double' class type, the values have been saved as a cell array with strings. Conversion to double can be easily achieved through the str2double command, at the cost of loss of numerical precision.
Examples
% Pascal Triangle for n <= 4
b = binomial(4);
b =
5×5 cell array
{'1'} {'0'} {'0'} {'0'} {'0'}
{'1'} {'1'} {'0'} {'0'} {'0'}
{'1'} {'2'} {'1'} {'0'} {'0'}
{'1'} {'3'} {'3'} {'1'} {'0'}
{'1'} {'4'} {'6'} {'4'} {'1'}
% Binomial expansion terms for n = 4
b = binomial(4, []);
b =
1×5 cell array
{'1'} {'4'} {'6'} {'4'} {'1'}
% Accurate N choose K (1000, 500)
b = round(vpa(binomial(1000, 500)));
b =
270288240945436569515614693625980000000000000092092817939260465262396897314854756510301009414481288655741593673629116029259908377031297694048974886875144010566190142039764976194007777780262204642682840529729586602474213985385651649048525534270439972471247027710107116837053571858737469793700061118464
% Table of binomial coefficients
x = [3:3:12; 0,3,6,8];
b = binomial(x); % Equal to: binomial(x'), or binomial(x(:, 1), x(:, 2))
b =
4×1 cell array
{'1' }
{'20' }
{'84' }
{'495'}
% Some binomial elements for a given 'n'
b = binomial(10, 0 : 2 : 10);
b =
1×6 cell array
{'1'} {'45'} {'210'} {'210'} {'45'} {'1'}
% Overflow and loss of numerical precision in 'double' types
b = binomial(60, 30);
err = str2double(b) - nchoosek(60, 30);
b = binomial(65, 35);
res = sym(b) - str2double(b);
err =
16
res =
-32

### Cite As

Moreno, M. (2023). Accurate Binomial Coefficients (https://www.mathworks.com/matlabcentral/fileexchange/109945-accurate-binomial-coefficients), MATLAB Central File Exchange. Retrieved .

##### MATLAB Release Compatibility
Created with R2022a
Compatible with any release
##### Platform Compatibility
Windows macOS Linux

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Version Published Release Notes
1.0.2

Increased range until IEEE 754 arithmetic output for class 'double' (n = 1029)

1.0.1

Increased the number of significant digits (before, it was set to '32').

1.0.0