1.

degree

2.

sim3coll <- function(nreps) 
{
   countColl <- 0
   count3 <- 0
   for (rep in 1:nreps) {
      tmp <- doExpt()
      nSend <- sum(tmp)
      if (nSend >= 2) {
         countColl <- countColl + 1
         if (nSend == 3)
            count3 <- count3 + 1
      }
   }
   count3 / countColl
} 

doExpt <- function() 
{
   trySend <- sample(0:1,3,replace=T,prob=c(0.6,0.4))
}

3.

Let Si denote the event that user i sends in that epoch, i = 1,2,3.
Then

P(S1 and S2 and S3 | S1 and S2 and not S3 or S1 and not S2 and S3 or ...)

= 0.4^3 / ( 3 * (0.4)^2 * 0.6 + 0.4^3)

4. 

# find largest gap

maxgap <- function(n,m,k) {
   tmp <- combn(n,m,checkgap)
   mean(tmp == k)
}

checkgap <- function(cmb) {
    tmp <- diff(c(cmb))
    max(tmp)
}