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version (2.81 KB) by Jos (10584)
Ratio of factorials, as in (a!b!c!...) / (d!e!f!g!...) (version 2.0, oct 2012)

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Updated 25 Oct 2012

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FACTORIALRATIO - ratio of factorial numbers

FR = FACTORIALRATIO (A,B) calculates the ratio between products of
factorials specified by A and B. A and B are lists of positive
integers, with N and M elements, respectively. The number of values in
each list may differ. The factorial ratio is based on the following formula:

(A1! * A2! * ... * AN!)
FR = -------------------------
(B1! * B2! * ... * BM!)

However, an approach is taken that increases the accuracy when A or B
contain larger values, as mentioned in the help of FACTORIAL.


f1 = factorialratio([10 8 6],[9 6 5]) ;
f2 = 10 * 8 * 7 * 6 ; % by hand
f3 = (factorial(10) * factorial(8) * factorial(6)) ./ ...
(factorial(9) * factorial(6) * factorial(5)) ;
disp('f1, f2, f3 = ') ; disp([f1 f2 f3])

% Also for larger values FACTORIALRATIO works correctly
f1 = factorialratio(150,143) ;
f2 = prod(144:150) ; % by hand
% where FACTORIAL lacks precision
f3 = factorial(150) ./ factorial(143) ;
disp('f1, f2, f3 = ') ; disp([f1 f2 f3])
disp(' Differences with f2 = ') ; disp([f1-f2 f2-f2 f3-f2])

% and even for extreme numbers things turn out well
f1 = factorialratio(30000,29998) ;
f2 = (30000 * 29999) ; % by hand
% whereas calls to FACTORIAL fail miserably ...
f3 = factorial(30000) ./ factorial(29998) ;
disp('f1, f2, f3 = ') ; disp([f1 f2 f3])

Note that, for instance, factorialratio(200,1) also fails, of course.
For such occasions other engines have to be used.

See also prod, factorial, gammaln

Cite As

Jos (10584) (2021). FACTORIALRATIO (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (4)

Jos (10584)

Version 2.0 (oct 2012) fixed the bug mentioned by Charles.


Version 1.1 (feb 2009) of this code has a bug.

x(1)=factorialratio(2,1); %should be 2
x(2)=factorialratio(1,2) %should be .5

x =

1 1

John D'Errico

Sorry, I forgot to point out what I feel to be one of the best features of this code - the copious internal comments that explain how it works.

I've often argued that the best MATLAB code should aim for a 1-1 ratio of comments to lines of code. That is, each line of code should have an associated comment line, at least in those places where the code is doing something tricky. This code goes beyond my aim. People interested in how to write good comments should look at what the author did here and learn from it.

John D'Errico

Good help as you would expect from Jos. Well written. An interesting solution for the computation itself.

MATLAB Release Compatibility
Created with R13
Compatible with any release
Platform Compatibility
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