It is ok to post!
# Name:Treasa Bency Biju Jose
# Date: 10-22-2020
# Assignment #13
-------------------------------------------------------------------------------------------------------------
1. [t=1, b=2, r=3, e=4, a=5, s=6, i=7, j=8, u=9]


G := [{4, 5, 7, 9}, {4, 5, 7, 9}, {4, 5, 7, 9}, {seq(i, i = 1 .. 9)}, {seq(i, i = 1 .. 9)}, {4, 5, 7, 9}, {seq(i, i = 1 .. 9)}, {4, 5, 7, 9}, {seq(i, i = 1 .. 9)}];
  G := [{4, 5, 7, 9}, {4, 5, 7, 9}, {4, 5, 7, 9}, 

    {1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, 

    {4, 5, 7, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, {4, 5, 7, 9}, 

    {1, 2, 3, 4, 5, 6, 7, 8, 9}]


f := normal(add(convert(GFt(G, i, t), `+`), i = 1 .. 9));
                               20 t + 9    
                     f := - ---------------
                                2          
                            20 t  + 4 t - 1

t := coeff(taylor(f, t = 0, 101), t, 100);
t := 63335077775392456206374190028134119432797969570629285269563\

  18314200638559392192528384

(i)

evalf(t/5^100);
                                      14
                        8.028674936 10  

(ii)

evalf(t/4^100);
                                      24
                        3.941351566 10  

(iii)

GFtx := proc(G, v, x, t) local n, eq, var, X, u, Neighs, eq1, nei, q, p; 
n := nops(G); 
var := {seq(X[u], u = 1 .. n)}; 
eq := {}; 
for u to n do 
  Neighs := G[u]; 
  if u = v then 
    eq1 := X[u] = 4 + t*x[u]*add(X[nei], nei in Neighs); 
  else 
    eq1 := X[u] = t*x[u]*add(X[nei], nei in Neighs); 
  end if; eq := eq union {eq1}; 
end do; 
var := solve(eq, var); 
factor(subs(var, [seq(X[u], u = 1 .. n)])); 
end proc;

F := normal(add(convert(GFtx([{2}, {1, 2}], i, x, t), `+`), i = 1 .. 2));
                            4 (t x[1] + 2)      
                F := - -------------------------
                        2                       
                       t  x[1] x[2] + t x[2] - 1

simplify(subs({x[1] = 5, x[2] = 4}, F));
                           -20 t - 8   
                        ---------------
                            2          
                        20 t  + 4 t - 1

-----------------------------------------------------------------------------------------------------------------------
2.
Wolf = 1, Farmer = 2, Sheep = 3, Cabbage = 4, Boat = 5


L := [{[], [1, 2, 3, 4, 5]}, {[4], [2, 1, 3, 5]}, {[1], [2, 3, 4, 5]}, {[3], [2, 1, 4, 5]}, {[1, 4], [2, 3, 5]}, 

     {[1, 4], [5, 2, 3]}, {[3], [1, 2, 4, 5]}, {[1], [2, 3, 4, 5]}, {[4], [1, 2, 3, 5]}, {[], [1, 2, 3, 4, 5]}];

seq(Paths(L, [[1, 2, 3, 4, 5], []], [[], [1, 2, 3, 4, 5]], i), i = 1 .. 10);
             {}, {}, {}, {}, {}, {}, {}, {}, {}, {}

-----------------------------------------------------------------------------------------------------------------------
4. 
Cannibal:=2; Missionaries:=1

P := [{[], [1, 1, 1, 3, 3, 3]}, {[3], [1, 1, 1, 3, 3]}, {[1, 2], [1, 1, 2, 2]}, {[1, 1, 1], [2, 2, 2]}, 

    {[2, 1], [1, 1, 2, 2]}, {[], [1, 1, 1, 2, 2, 2]}];


seq(Paths(L, [[1, 1, 1, 2, 2, 2], []], [[], [1, 1, 1, 2, 2, 2]], i), i = 1 .. 10);
             {}, {}, {}, {}, {}, {}, {}, {}, {}, {}

-----------------------------------------------------------------------------------------------------------------------