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Tuples[list,n]

generates a list of all possible n‐tuples of elements from list.

Tuples[{list1,list2,…}]

generates a list of all possible tuples whose i th element is from listi.

Tuples[list,{n1,n2,…}]

generates a list of all possible n1n2… arrays of elements in list.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Generalizations & Extensions  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
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History
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Tuples

Tuples[list,n]

generates a list of all possible n‐tuples of elements from list.

Tuples[{list1,list2,…}]

generates a list of all possible tuples whose i th element is from listi.

Tuples[list,{n1,n2,…}]

generates a list of all possible n1n2… arrays of elements in list.

Details

  • The elements of list are treated as distinct, so that Tuples[list,n] for a list of length k gives output of length kn.
  • Tuples[list,n] is equivalent to Tuples[Table[list,n]].
  • The order of elements in Tuples[list,n] is based on the order of elements in list, so that Tuples[{a1,…,ak},n] gives {{a_(1),a_(1),...,a_(1)},{a_(1),a_(1),...,a_(2)},...,{a_(k),a_(k),...,a_(k)}}.
  • The object list need not have head List. The head at each level in the arrays generated by Tuples will be the same as the head of list.

Examples

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Basic Examples  (3)

All possible 3-tuples of 0 and 1:

The first element given is always picked first:

All ways of picking one element from each list:

Generalizations & Extensions  (3)

Each occurrence of an element is picked separately:

All possible 2×2 arrays formed from a and b:

Tuples with any head:

Applications  (8)

Digits of successive base-2 numbers:

All possible trigrams of A and B:

Neighborhood rules for an elementary cellular automaton:

Digit sequences, or evolution of a substitution system:

2D lattice of points:

Connect pairs of points in all possible ways:

Properties & Relations  (4)

Tuples[list,n] is equivalent to Tuples[Table[list,n]]:

All possible digit sequences:

Results from Outer involve extra sublists:

Subsets picks each element at most once, and treats reorderings as equivalent:

Possible Issues  (1)

With one argument, Tuples picks an element from each sublist:

With two arguments, Tuples treats each sublist as a complete element:

Neat Examples  (6)

3D lattice of points:

Wolfram Research (2004), Tuples, Wolfram Language function, https://reference.wolfram.com/language/ref/Tuples.html.

Text

Wolfram Research (2004), Tuples, Wolfram Language function, https://reference.wolfram.com/language/ref/Tuples.html.

CMS

Wolfram Language. 2004. "Tuples." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Tuples.html.

APA

Wolfram Language. (2004). Tuples. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Tuples.html

BibTeX

@misc{reference.wolfram_2025_tuples, author="Wolfram Research", title="{Tuples}", year="2004", howpublished="\url{https://reference.wolfram.com/language/ref/Tuples.html}", note=[Accessed: 11-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_tuples, organization={Wolfram Research}, title={Tuples}, year={2004}, url={https://reference.wolfram.com/language/ref/Tuples.html}, note=[Accessed: 11-August-2025]}