NCHOOSE  all combinations of the elements of a set
W = nchoose(S) returns all possible combinations of one or more
elements of the set S. In total there are 2^N1 combinations, where N
is the number of elements in S. W is a cell array: each cell holds one
of these combination (as a row vector). S can be a cell array of
strings, and each cell of W will then contain a cell array of strings.
Example:
nchoose([2 4 6 8])
% > { [2] ;
% [4] ;
% [2 4] ;
% [6] ;
% ...
% [2 6 8] ;
% [4 6 8] ;
% [2 4 6 8]} ; % in total (2^4)1 = 15 different combinations
Notes:
 For sets containing more than 18 elements a warning is given, as this
can take some time. On my PC, a set of 20 elements took 20 seconds. Hit CtrlC to intterupt calculations.
 If S contain nonunique elements (e.g. S = [1 1 2]), NCHOOSE will return nonunique 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.
 By adding an empty cell to the output, the power set of a cell array
is formed:
S = {1 ; 'hello' ; { 1 2 3}}
PowerS = nchoose(S)
PowerS(end+1) = {[]}
See: http://en.wikipedia.org/wiki/Power_set
See also NCHOOSEK, PERMS
and PERMN, ALLCOMB, NCHOOSECRIT on the File Exchange
2.2  updated description 

2.2  update contact info. Works on most ML releases 

1.1  reference to powerset 

(v1.1) fixed spelling errors 
Inspired: NCHOOSECRIT (V1.0 feb 2013)
Marc Lalancette (view profile)
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 (view profile)
Joshua Carmichael (view profile)
Vectorized, and just what I need. Very fast.
as had to be expected from this author: exemplary MLtypical implementation of an evergreenrequest on CSSM with good help and copious comments for its sleek computational engine.
this is a true addon 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
Very useful thoughts indeed. Thanks us! The update (v2.0) incorporates the (slightly altered) bitget implementation.
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(1numel(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.^(nb1:1:0).';
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
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.