Proving that these are all possible permutations

In summary, the conversation discusses the six possible permutations when arranging the numbers 1, 2, and 3 in different orders. It also mentions the possibility of rigorously proving these arrangements and the use of combinatoric identities.
  • #1
maxbashi
18
0
So if I have numbers 1,2,3, intuitively you can say that there are 6 permutations:

123
132
213
231
312
321

If there any way to rigorously prove that these are the only possible arrangements of these three numbers? Even more simple, is there any way to prove that 12 and 21 are the only permutations of 1 and 2? I don't need a ton of detail if it's going to be tedious, I'm just curious.

Thanks
 
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  • #2
There are only 3! = 6 possible arrangements (three possibilities for the first position, two possibilities for the second, and one for the third), and you've listed six different arrangement. Thus, you listed all possible arrangements.
 
  • #3
Hey maxbashi and welcome to the forums.

There is actually a set of formal definitions for permutations that you could use and would be helpful if you had to do more general or abstract permutations but for this specific problem it might be wise to use Number Nines advice and use the combinatoric identity for the number of unrestricted permutations (i.e. 3! = 6).
 

FAQ: Proving that these are all possible permutations

What does it mean to prove that something is a permutation?

A permutation is a rearrangement of a set of elements in a specific order. Proving that something is a permutation means showing that all possible ways of arranging those elements have been considered and accounted for.

How do you prove that all permutations have been accounted for?

To prove that all permutations have been accounted for, you must demonstrate that every possible combination of elements has been included in the arrangement. This can be done by systematically listing out each permutation or by using mathematical formulas to calculate the total number of possible permutations.

What evidence is needed to prove that something is a permutation?

The evidence needed to prove that something is a permutation will depend on the context and the specific problem being addressed. However, in general, a clear and logical explanation of how all permutations have been considered and accounted for will be required.

Can you use a computer to prove that something is a permutation?

Yes, a computer can be used to generate and verify all possible permutations of a set of elements. However, it is important to ensure that the algorithm used is accurate and accounts for all possible combinations in order to be considered a valid proof.

Why is proving that something is a permutation important?

Proving that something is a permutation is important because it provides a systematic and logical approach to problem-solving. It also ensures that all possible solutions have been considered and accounted for, increasing the accuracy and reliability of the results.

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