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## nchoose

version 3.0 (2.87 KB) by
all combinations of the elements of a set

Updated 08 Feb 2019

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W = nchoose(S) returns all possible combinations of 0, 1, or more
elements of the set S, having N elements. There are 2^N combinations
in total. W is a cell array and each cell holds one of these
combination (as a row vector).
S can be a cell array, and each cell of W will then contain a cell
array. W is the powerset of S, as it includes the empty set (0
elements) as it first cell.

For a vector of integers I, W = nchoose(S, I) returns only the sets
indicated by the indices I. This might be useful for large sets.

Examples:
nchoose([2 4 6 8])
% -> { [] ;
% [2] ;
% [4] ;
% [2 4] ;
% [6] ;
% ...
% [2 6 8] ;
% [4 6 8] ;
% [2 4 6 8]} ; % in total 16 different combinations

nchoose([33 22 11], [1 8 4])
% -> { [] ; [33 22 11] ; [ 33 11]}

Notes:
- For sets containing more than 18 elements a warning is given, as this
can take some time. Hit Ctrl-C to intterupt calculations.
- If S contain non-unique elements (e.g. S = [1 1 2]), nchoose will
return non-unique cells. In other words, nchoose treats all elements
of S as being unique. One could use nchoose(UNIQUE(S)) to avoid that.
- Loosely speaking, nchoose(S) collects all output of multiple calls to
NCHOOSEK(S, K) where K is looping from 1 to the number of elements of
S. The implementation of nchoose, however, does rely of a different
method and is much faster than such a loop.

permn, nchoose2, allcomb on the file Exchange

### Cite As

Jos (10584) (2020). nchoose (https://www.mathworks.com/matlabcentral/fileexchange/20011-nchoose), MATLAB Central File Exchange. Retrieved .

Marc Lalancette

Theoretical note, in case someone (as I did) wants to get a limited number of random combinations, then this method fails for N > 51 since then eps(2^52) = 1, i.e. can't represent the integers accurately any more with double type. Of course, here we run out of memory with such a large N anyway.

Also, with bitget, I was using a loop over bits instead of a loop over combinations. I haven't checked, but I assumed it would be faster. I'm also going to check with d2b instead of bitget. (found on FEX, and modified to work on arrays).

Eric

Joshua Carmichael

Vectorized, and just what I need. Very fast.

Urs (us) Schwarz

as had to be expected from this author: exemplary ML-typical implementation of an evergreen-request on CSSM with good help and copious comments for its sleek computational engine.
this is a true add-on to the family of (basic) ML functions like dec2bin - and one wonders why it has not been supplied by TMW from the very beginning (or at leas since the implementation of CELLs) because users from many different fields will be able to use it.
us

Jos (the author)

Very useful thoughts indeed. Thanks us! The update (v2.0) incorporates the (slightly altered) bitget implementation.

Urs (us) Schwarz

as usual, interesting stuff, jos.
a few remarks:
- it should be mentioned that UNIQUE refers to the position uniqueness as something like
>> nchoose([1,1,1]);
will yield (seemingly) equal combinations
- there should be an error check for M > 23
- a speed improvement
W=cell(M,1);
p2=pow2(1-numel(S):0);
for i=1:M,
Q=rem(floor(i * p2),2)==1 ;
W{i}=S(Q); % take consuming FLIPLR out!
end
W=cellfun(@(x) x(end:-1:1),W,'uni',false);

i used this approach in the past, which yields a speed gain of approximately 2.6

% note: all checks/comments take out...
function cmb=pcomb(pat)
nb=numel(pat);
mc=(2.^nb)-1;
pp=2.^(nb-1:-1:0).';
cmb=cell(mc,1);
for i=1:mc
cmb{i,1}=pat(bitget(i*pp,nb)~=0);
end
end

comparison on a wintel:
ic2.2*2.4mhz/2gb/winxp.sp2/r2007b

len: ...pcomb | .nchoose | ....gain
01: 0.000040 | 0.000107 | 3.202227
02: 0.000043 | 0.000119 | 3.228383
03: 0.000046 | 0.000148 | 3.302875
04: 0.000075 | 0.000238 | 3.082459
05: 0.000127 | 0.000387 | 3.065482
06: 0.000232 | 0.000714 | 3.120286
07: 0.000447 | 0.001342 | 2.991717
08: 0.000908 | 0.002625 | 2.878488
09: 0.001813 | 0.005216 | 2.881585
10: 0.003804 | 0.010127 | 2.655678
11: 0.007689 | 0.020627 | 2.682076
12: 0.015583 | 0.041467 | 2.657087
13: 0.031492 | 0.082831 | 2.624573
14: 0.064452 | 0.168786 | 2.617609
15: 0.130970 | 0.341670 | 2.607633
16: 0.268776 | 0.688592 | 2.560370
17: 0.585902 | 1.393225 | 2.379455
18: 1.285114 | 2.812605 | 2.188545
19: 2.782169 | 5.672041 | 2.039558
20: 6.063693 | 11.488511 | 1.894641
21: 13.237517 | 22.986104 | 1.736436
22: 28.864961 | 46.540510 | 1.612353

just a few thoughts
us

John D'Errico

Ok, I'll admit that I'd have expected this would already have been on the FEX or in Matlab. And for myself, I'd just use the simple dec2bin(1:(2^n))=='1'. After all, a set is defined in Matlab only by your representation of it. Numbers are merely symbols, and symbols have meaning only in context.

On the other hand, this is nicely written. It takes the dec2bin solution one step further, so it could clearly be of use to others. It has good help, with an example.

The author has taken great pains in the code to document why he chose the method used. Its a style I'd highly recommend for others. In some future time when you need to revisit your code, you will very much appreciate these notes. Or if someone else wants to read your code, they will wonder why you chose to solve the problem as you did. These comments answer all of those questions. Well done.

##### MATLAB Release Compatibility
Created with R2018a
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