> #Attendence Q1.
;
> #Describe the problem that Euler solved regardinig 7 bridges. 
;
> #There are (or were, in Euler’s time) seven bridges to connect the two islands and the downstream parts of the town. 
;
> #Euler wondered if a person could walk across each of the seven bridges once and only once to touch every part of the town. Starting and ending at the same spot was not a requirement
;
# 
> 
;
> 
;
> 
;
> #Attendence Q2
;
> #Draw it on a piece of paper with line segment representing every edge
;
> #G=[{2,3}, {1,3,4},{1,2,4},{2,3}]; 
> #V={1,2,3,4}
;
> 
;
# 
> #Attendence Q3
;
> #If toss a fair coin 2000 times, what is the probability that you get the exact 1000 heads (and 1000 tails) 
;
> with(combinat)
[Chi, bell, binomial, cartprod, character, choose, composition, conjpart, 
decodepart, encodepart, eulerian1, eulerian2, fibonacci, firstcomb, firstpart,
firstperm, graycode, inttovec, lastcomb, lastpart, lastperm, multinomial, 
nextcomb, nextpart, nextperm, numbcomb, numbcomp, numbpart, numbperm, partition
, permute, powerset, prevcomb, prevpart, prevperm, randcomb, randpart, randperm
, rankcomb, rankperm, setpartition, stirling1, stirling2, subsets, unrankcomb,
unrankperm, vectoint]

;
> a:=binomial(2000,1000);
Typesetting:-mprintslash([(a := 20481516269894897143351625029808250443964248879\
81397033820382637671748186202083755828932994)],[2048151626989489714335162502980\
825044396424887981397033820382637671748186202083755828932994])

;
> b:=2^2000;
Typesetting:-mprintslash([(b := 11481306952742545242328332011776819840223177020\
8869520047764273682576626139237031385665948631)],[11481306952742545242328332011\
7768198402231770208869520047764273682576626139237031385665948631])

;
> evalf(a/b)
.1783901115e-1

;
> 
;
> 
;
> #Attdence Q4
;
> #If you toss a dice 6000 times, what is the probability of that each of posssible outccomes {1,2,3,4,5,6}
;
> #p(1)=p(2)=p(3)=p(4)=p(5)=p(6)=1/6
;
> 
;
> 
;
> 
;
> #Attendence Q5
;
> #Pick 5 random facebook friends
;
> #for each of them pick 3 friends
;
> #for each of friends pick 3 friends 
;
> #Label the people icked 1,2,...,n
;
> #write the graph in data structure
;
> #V:={Tom, Jake, Alissa, Bob, Lucy};
> #G:=[{Jake,Alissa,Bob}, {Lucy, Alissam Tom},{Tom, Jake, Bob}, {Jakem Tom, Lucy}, {Jake, Aliisa, Bob}];
> 
;
> 
;
> #Attendence Q6
;
> #Look at the cities that border Piscataway, and for each of them those that border them, 
;
> #and again until you get to princton
;
> #(i) construct the graph
;
> #(ii) Find the path from piscataway to princiton 
;
> #(a) find the actual path using Paths( and list them)
;
> #(b) by using NuPathro
;
> #(c) by using GFt
;
> 
;
> #South Plainfield:=1 Edison:=2, North Brunswick: =3, South Brunswick:=4, New Brunswick:=5, Franklin: 6, Princton:=7, Piscataway: 8
;
> #V:={1, 2, 3, 4, 5, 6, 7, 8,}
;
> G:=[{8, 2}, {8, 1, 5}, {5, 6, 4}, {7, 6, 3},{8, 2, 3, 6}, {8, 5, 3, 4, 7}, {6, 4},{1, 2, 5, 6}];
Typesetting:-mprintslash([(G := [{2, 8}, {1, 5, 8}, {4, 5, 6}, {3, 6, 7}, {2, 3
, 6, 8}, {3, 4, 5, 7, 8}, {4, 6}, {1, 2, 5, 6}])],[[{2, 8}, {1, 5, 8}, {4, 5, 6
}, {3, 6, 7}, {2, 3, 6, 8}, {3, 4, 5, 7, 8}, {4, 6}, {1, 2, 5, 6}]])

;
> Paths:=proc(G,u,v,k)  local Neighs, PA,nei,PA1,pa1:
> option remember:
> 
> #INITIAL CONDITIONS, paths of length 0 are just lists of length 1
> if k=0 then
>  if u=v then
>    RETURN({[u]}):
>   else
>    RETURN({}):
>  fi:
> fi:
> 
> #Every path from u to v of length k starts with a first step that is towards one of the neighbors of u
> #so let's find the set of neighbors of u
> Neighs:=G[u]:
> 
> #We will dynamically construct all the paths from u to v of length k by 
> #going through all the options for first step
> #PA is the desired output
> #We start out PA the empty set
> PA:={}:
> 
> for nei in Neighs do
>  #We call the procedure recursively to find the set of paths from a typical neighbor of u, to v of length one less, i.e. k-1
>  PA1:=Paths(G,nei,v,k-1):
> 
>  #For each of these paths we stick u at the beginning and add these to our desired output
> PA:=PA union {seq([u,op(pa1)], pa1 in PA1)}:
> od:
> 
> PA:
> 
> end:
> Paths(G, 8,7 , 1)
{}

;
> Paths(G,8,7,2)
{[8, 6, 7]}

;
> Paths(G,8,7,3)
{[8, 5, 6, 7], [8, 6, 4, 7]}

;
> Paths(G,8,7,4)
{[8, 1, 8, 6, 7], [8, 2, 5, 6, 7], [8, 2, 8, 6, 7], [8, 5, 3, 4, 7], [8, 5, 3,
6, 7], [8, 5, 6, 4, 7], [8, 5, 8, 6, 7], [8, 6, 3, 4, 7], [8, 6, 3, 6, 7], [8,
6, 4, 6, 7], [8, 6, 5, 6, 7], [8, 6, 7, 4, 7], [8, 6, 7, 6, 7], [8, 6, 8, 6, 7]
}

;
> Paths(G,8,7,5)
{[8, 1, 2, 5, 6, 7], [8, 1, 2, 8, 6, 7], [8, 1, 8, 5, 6, 7], [8, 1, 8, 6, 4, 7]
, [8, 2, 1, 8, 6, 7], [8, 2, 5, 3, 4, 7], [8, 2, 5, 3, 6, 7], [8, 2, 5, 6, 4, 7
], [8, 2, 5, 8, 6, 7], [8, 2, 8, 5, 6, 7], [8, 2, 8, 6, 4, 7], [8, 5, 2, 5, 6,
7], [8, 5, 2, 8, 6, 7], [8, 5, 3, 4, 6, 7], [8, 5, 3, 5, 6, 7], [8, 5, 3, 6, 4,
7], [8, 5, 6, 3, 4, 7], [8, 5, 6, 3, 6, 7], [8, 5, 6, 4, 6, 7], [8, 5, 6, 5, 6,
7], [8, 5, 6, 7, 4, 7], [8, 5, 6, 7, 6, 7], [8, 5, 6, 8, 6, 7], [8, 5, 8, 5, 6,
7], [8, 5, 8, 6, 4, 7], [8, 6, 3, 4, 6, 7], [8, 6, 3, 5, 6, 7], [8, 6, 3, 6, 4,
7], [8, 6, 4, 3, 4, 7], [8, 6, 4, 3, 6, 7], [8, 6, 4, 6, 4, 7], [8, 6, 4, 7, 4,
7], [8, 6, 4, 7, 6, 7], [8, 6, 5, 3, 4, 7], [8, 6, 5, 3, 6, 7], [8, 6, 5, 6, 4,
7], [8, 6, 5, 8, 6, 7], [8, 6, 7, 4, 6, 7], [8, 6, 7, 6, 4, 7], [8, 6, 8, 5, 6,
7], [8, 6, 8, 6, 4, 7]}

;
> #for step length = 6,7,8 there are too many paths. 
;
> 
;
> #By using NuPaths()
;
> NuPaths:=proc(G,u,v,k)  local Neighs,nei:
> option remember:
> 
> #INITIAL CONDITIONS, paths of length 0 are just lists of length 1
> if k=0 then
>  if u=v then
>    RETURN(1):
>   else
>    RETURN(0):
>  fi:
> fi:
> 
> #Every path from u to v of length k starts with a first step that is towards one of the neighbors of u
> #so let's find the set of neighbors of u
> Neighs:=G[u]:
> 
> #The number of paths of length k from u to v is the sum of the number of paths from each neighbor of u to v, of length k-1
> 
> add(NuPaths(G,nei,v,k-1),nei in Neighs):
> 
> end:
> 
;
> NuPaths(G,8,7,2)
1

;
> NuPaths(G,8,7,3)
2

;
> NuPaths(G,8,7,4)
14

;
> NuPaths(G,8,7,5)
41

;
> NuPaths(G,8,7,6)
174

;
> #Now, using GFT
;
> GFt:=proc(G,v,t)  local n,eq,var,X,u,Neighs,eq1,nei:
> 
> n:=nops(G):
> 
> #var is the set of desired quantities. The outputs is going to be [X[1], ..., X[n]] after it is SOLVED
> var:={seq(X[u],u=1..n)}:
> 
> #We now use the powerful technique of weight-enumerators to set a SYSTEM of equations for these quantities
> #The weight enumerator according to the weight t^LengthOfPath of all paths from u to v 
> #is t times the sum of the weight enumerators of all path from neighbors of u and if u=v you have to add
> #1 for the weight of the empty path
> eq:={}:
> 
> 
> #Neig is the set of neighbors of u
> for u from 1 to n do
>  Neighs:=G[u]:
>  if u=v then
>   eq1:=X[u]=1+t*add(X[nei],nei in Neighs):
>  else
>   eq1:=X[u]=t*add(X[nei],nei in Neighs):
>  fi:
> eq:=eq union {eq1}:
> od:
> 
> 
> 
> #We now let Maple solve the linear system of equations
> var:=solve(eq,var):
> 
> #we now plug the solution into seq vector of unknowns [X[1], ..., X[n]]:
> 
> factor(subs(var,[seq(X[u], u=1..n)])):
> 
> end:
> 
;
> normal(add(convert(GFt(G,i,t),`+`), i=1..8))
-2*(3*t^7+18*t^6+10*t^5-47*t^4-56*t^3-6*t^2+13*t+4)/(3*t^8+12*t^7-7*t^6-52*t^5-\
35*t^4+12*t^3+13*t^2-1)

;
> 
;