What is the notation to describe the following problem:
Using only the numbers 3,4,7 at the same time, with no repetition, write all the possible numbers. Like, $$374,473,437,473...$$
I tried to make $$ A = \{3,4,7\} f: x,y,z \rightarrow A, (x,y,z) \rightarrow 100x+10y+z $$ But what i did is possibly wrong and incomplete.
may i please u give other ideas.
thanks folks
A permutation of a set is an ordered arrangement of its elements. For example, the permutations of $\{1,2,3\}$ are: $123,132,213,231,312,321$
A set cannot include repeated elements, so a multiset is needed if you want to find all numbers that can be formed by a list that includes repeats, e.g., $3,3,4,$ and $5$.
This is actually an extremely common problem in combinatorics, although it's more commonly looked at in terms of letters than numbers. This site has many examples of such problems being asked (usually asking for the number of total permutations rather than a list of them).
In terms of group,s this can be expressed as $S_{|M|}$ for the case of there being no repeats. When there are repeats, it would be $S_{|M|}/Sym_{M}$. Here $M$ is the (multi)set you are interested in the permutations of (in the example, $M=\{3,4,7\}$), $S_k$ is the symmetric group on $k$ elements, and $Sym(M)$ is the set of symmetries of $M$, i.e. the set that tells us how many of the permutations yield the same result. This is only relevent if there is repeats in the string of symbols, such as $M=\{3,3,4,7\}$. In that case our expression gives $S_4/S_2\simeq A_4$. $A_4$ (the alternating group on four elements) is the name of a particular permutation group that is the answer to the problem for the case of $M=\{3,3,47\}$