# how to pick up all combination of numbers from multiple vectors

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mohamed Faraj on 13 Aug 2019
Edited: Adam Danz on 4 Apr 2023
I have a number of vectors, probably with different lengths, e.g., a=[1 2 3], b=[4 5 6 7] and c=[8 9 10 11 12]. From a, b and c I have 3*4*5=60 possible points, e.g., one possibility is (1,4,8). If I know the number of vectors and the length of each vector in advance, this is easy to program. However, I want to write a general code that can find all these combinations regardless of the number of vectors and their individual lengths
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Mike Croucher on 4 Apr 2023

John D'Errico on 13 Aug 2019
Don't store your vectors separately. Instead, learn to use tools like cell arrays, which make things hugely more efficient.
V = {[1 2 3],[4 5 6 7],[8 9 10 11 12]};
Next, how do you use a cell array for this purpose? You pass the elements into ndgrid, using what is called acomma separated list. (Or meshgrid.) That is what you get when you type V{:}, a comma separated list. It allows you to pass in each element of the cell array into a function as if each element of the cell array was an argument of the function.
For example, if we did this:
[G1,G2,G3] = ndgrid(V{:});
Hmm. That creates three different arrays, that do contain all combinations of the elements of those vectors if you look carefully. But here we don't want to split the results into n different named arrays. We want a cell array as output. So now, try this:
C = cell(1,numel(V));
[C{:}] = ndgrid(V{:})
C =
1×3 cell array
{3×4×5 double} {3×4×5 double} {3×4×5 double}
Better. We have captured the output from ndgrid back into a cell array. But what we probably wanted was one flat array, with three columns, and here, 60 rows. We could convert each of those arrays into a column vector easily enough.
C = cellfun(@(X) reshape(X,[],1),C,'UniformOutput',false)
C =
1×3 cell array
{60×1 double} {60×1 double} {60×1 double}
And, now finally, just convert C into a flat array, using a tool like horzcat. (square brackets will suffice. That is...
C = horzcat(C{:})
C =
1 4 8
2 4 8
3 4 8
1 5 8
2 5 8
3 5 8
1 6 8
2 6 8
3 6 8
1 7 8
2 7 8
3 7 8
1 4 9
2 4 9
3 4 9
1 5 9
...
2 7 12
3 7 12
As you should see, nothing I did was dependent on the size of the arrays, the length of the vectors, the number of different vectors. That was all driven by the one initial cell array as I created it. Learn to use MATLAB, as it was designed to be used.
진환 유 on 16 Mar 2022
it was really helpful for me. Thank you for good solution!

Adam Danz on 13 Aug 2019
Edited: Adam Danz on 4 Apr 2023
Here are two solutions for before/after R2023a.
Use combinations in MATLAB R2023a or later
% Demo data
a = 1:3;
b = 11:13;
c = -4:1;
d = 9;
allCombinations = combinations(a,b,c,d)
allCombinations = 54×4 table
a b c d _ __ __ _ 1 11 -4 9 1 11 -3 9 1 11 -2 9 1 11 -1 9 1 11 0 9 1 11 1 9 1 12 -4 9 1 12 -3 9 1 12 -2 9 1 12 -1 9 1 12 0 9 1 12 1 9 1 13 -4 9 1 13 -3 9 1 13 -2 9 1 13 -1 9
Before R2023a
This solution places all vectors in a cell array and uses ndgrid to create permutations. See comments for more detail. T
% Use the same demo data created above.
% Put all vectors into cell array
allVecs = {a,b,c,d};
sub = cell(1,numel(allVecs));
[sub{:}] = ndgrid(allVecs{:});
sub = cellfun(@(x)x(:),sub,'UniformOutput', false);
% allPerms is [m x n] matrix of m permutations of n vectors
% m should equal prod(cellfun(@numel,allVecs))
% n should equal numel(allVecs)
allPerms = cell2mat(sub)
allPerms = 54×4
1 11 -4 9 2 11 -4 9 3 11 -4 9 1 12 -4 9 2 12 -4 9 3 12 -4 9 1 13 -4 9 2 13 -4 9 3 13 -4 9 1 11 -3 9
Compare results
combinations() produces a table and my second solution produces a matrix with rows in a different order. Here I'll resort the rows of the table to match the rows of the matrix and we'll confirm that the two solutions are equal.
allCombinationsMat = sortrows(allCombinations{:,:},[4 3 2 1]);
isequal(allCombinationsMat, allPerms)
ans = logical
1
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Adam Danz on 14 Aug 2019
Sounds good. So your variable "x" is my variable "allVecs". Did you have any trouble implementing the rest of the solution?

Bruno Luong on 13 Aug 2019
Edited: Bruno Luong on 13 Aug 2019
a=[1 2 3], b=[4 5 6 7], c=[8 9 10 11 12]
C = {a,b,c}; % put you vectors here
n = length(C);
[C{:}] = ndgrid(C{:});
C = reshape(cat(n+1,C{:}),[],n)
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Bruno Luong on 16 Mar 2022
"could we limit sum of rows?"
Simply postfilter what you want to keep (such as limit of sum) afterward.

Chris on 14 Aug 2019
I use allcomb from the Exchange - works great. Works with chars too. It uses ndgrid under the hood and is probably mostly a packaged up version of the code other have shown here.