Similar to this question, [Frequency of Item in Combination](https://stats.stackexchange.com/questions/229044/frequency-of-item-in-combination). 

I am randomly sampling S objects out of N=99 objects into 9 boxes labeled by a single character, "A-I". **Question 1: I want to find the probability of M boxes having 2 or more objects, dependent on how many objects I sample, S out of N.** Currently I am only able to do so by simulation (see code below), which is RNG dependent, but accurate. **Question 2: Is there a distribution I can follow here?** I'm bouncing between binomial and hypergeometric, but I am uncertain how to implement it.

    mycountL <- double(9)
    names(mycountL) <- LETTERS[1:9]
    
    # Change this for sample drawing size
    xTimes = 18

    set.seed(12)
    for(i in 1:10000){
      nL <- names(which(table(sample(rep(LETTERS[1:9],11), xTimes ))>=2))
      lL <- length(nL)
      mycountL[lL] <- mycountL[lL]+1
    }
    
    mycountL/10000 #For probabilities. Drawing 18 times is the lowest sample possible to draw exactly 2 in each LETTER, except that it is highly unlikely.


### Edits: Clarification