Question/HMMT 2023 Guts Q6

There are five people in a room.They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them.Compute the probability that there exists a group of three people such that each of them is pointing at the other two in the group.

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List all results from exact enumeration

Choose two isolated people and they are free to point to any body. The rest three people point to each other:
In[]:=
wholeGrp={1,2,3,4,5};
In[]:=
isgCases=Subsets[wholeGrp,{2}]
Out[]=
{{1,2},{1,3},{1,4},{1,5},{2,3},{2,4},{2,5},{3,4},{3,5},{4,5}}
Use the combination of Complement and Subset function to dynamically change the boundaries for the second and third iterators based on which pair is chosen from the list above.:
In[]:=
edgeList=Flatten[Table[Join[​​{isg[[1]]p1[[1]],isg[[1]]p1[[2]],isg[[2]]p2[[1]],isg[[2]]p2[[2]]},​​DirectedEdge@@@Permutations[Complement[wholeGrp,isg],{2}]​​],​​{isg,isgCases},​​{p1,Subsets[Complement[wholeGrp,{isg[[1]]}],{2}]},​​{p2,Subsets[Complement[wholeGrp,{isg[[2]]}],{2}]}​​],2];
Double check that the enumeration is correct:
In[]:=
{edgeList//Length,Binomial[5,2]*Binomial[4,2]^2}
Out[]=
{360,360}
Thus the probability is
In[]:=
360/Binomial[4,2]^5
Out[]=
5
108

Graph visualization

Convert the edge list above into circular graphs with fixed seating for each person:
circleLayout[n_]:=Table[{Cos[2π/ni],Sin[2π/ni]},{i,0,n-1}];​​vrtxCrd=Thread[Range[5]->circleLayout[5]];
In[]:=
validG=(Graph[#,​​VertexCoordinates->vrtxCrd,​​VertexLabels->Placed["Name",Center],​​VertexLabelStyle->Cyan,​​VertexSize->.3])&/@edgeList;
The target subgraph or its isomorphism.The exact names of the vertices can be chosen arbitrarily:
In[]:=
h=Graph[DirectedEdge@@@Permutations[{1,2,3},{2}]]
Out[]=
Randomly choose the valid graph and highlight the subgraph:
In[]:=
SeedRandom["HMMT2023+WL"];​​GraphicsGrid@Map[With[{res=FindSubgraphIsomorphism[h,#]},​​HighlightGraph[#,​​VertexReplace[h,First[res//Normal]],​​GraphHighlightStyle->"DehighlightGray",​​ImageSize->140]​​]&,RandomChoice[validG,{3,3}],{2}]
Out[]=

CITE THIS NOTEBOOK

Work with Notebook Assistant on a HMMT probability problem​
by Shenghui Yang​
Wolfram Community, STAFF PICKS, December 15, 2024
​https://community.wolfram.com/groups/-/m/t/3339313