#OK to post homework
#Ariana Yousafzai, 11/22/2020, Assignment 21
1. 16530019558728142168379389104054542948954902506442870676877264555437768356978273337017649902196267779582098490170060


5. 
#EstProbKcomps(n,m,k,N): estimates the probability that a random graph with n vertices and m edges has exactly k components, by sampling N such random graphs.
#by simulating N times. 
EstProbConn:=proc(n,m,k,N) local i,co, G:
co:=0:
for i from 1 to N do
G:=RandGr(n,m):
if nops(CCs(G))=1 then
 co:=co+1:
fi:
od:
evalf(co/N):
end:
6. 
#ProbKcomponents(n,m,k): uses the method of exponential generating functions to find the EXACT probability that a random graph with n vertices and m edges has k components. Note that ProbKcomponents(n,m,1) is EXACTLY the same as ProbConn(m,m,1)
ProbConn:=proc(n,m,k)  local a:
evalf(coeff(WtEdConGclever(n,a)[n],a,m)/binomial(n*(n-1)/2,m)):
end