What is the best way to test of draws ( 10,000 unique possibilities) from multiple sample are random? say I have 6 people drawing from their own box, each box contains 10,000 unique barcodes.  Now at the end of the experiment each person has drawn roughly 10-20 barcodes. How do I test if the barcodes are not siginficantly overlapping between each person. In other words I want to show that the barcodes are just random? I'm thinking of something like a hypergeometric test or some sort of contigency table, however in this case there there are N=6 and 10,000 unique rows.
thanks!

R code for simulation per discussion in comments.
For comments I did 100,000 iterations; here I did
a million iterations for greater precision.
set.seed(806)   # for reproducibility
m = 10^6
d = numeric(m)  # duplicates
n = 80          # barcodes chosen
for (i in 1:m) {
 x = sample(1:10000, n, rep=T)
 d[i] = n - length(unique(x))
 }
mean(d)
[1] 0.315031   # mean nr of matches among 80
summary(d)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.000   0.000   0.000   0.315   1.000   6.000 

quantile(d, c(.9, .95, .99))
90% 95% 99% 
  1   1   2    # rare to get 2 or more matches

With n = 20, we see that it is rare to have one duplicate in a single group of 20. [Even more so for a single group of 10.]
