One of my friends is building a game where the player will get questions from 6 different categories. Each category has a total of 50 questions. A single game consists of answering one question from each category. Questions are selected randomly and independent of previous games. In each game the drawings from each category are also independent.

From observations we have concluded that 1) either there's a high probability for a player to get the same questions multiple times when e.g. 20 games have been played. 2) or our random function is not random at all as people do get the same questions regularly.

I have a basic understanding of probability theory and have concluded that we are dealing with multiple binomial distributions (the categories) where the probability for a particular event to occur is p=0.02

If we were dealing with only one category there are many online calculators to put in relevant values for N, k and p http://www.vassarstats.net/textbook/ch5apx.html.

I'm just not sure how to deal with the multiple categories we have.

Does the rule for sums of binomials apply? http://en.wikipedia.org/wiki/Binomial_distribution#Sums_of_binomials