TITLE: Algorithm creating subsets with certain properties
QUESTION [5 upvotes]: I'm trying so solve following problem:
Let's say, we have a set $A=\{1,2,3,...,49\}$.
Now, I am defining sets $A_1, A_2, A_3,...,A_n$ as follow: $A_1=\{a_1,a_2,a_3,...,a_{30}\}$,  $A_2= \{b_1,b_2,b_3,...b_{30}\}$, and so on, where all elements of sets $A_i$ are also elements of set $A$, which means they are subsets of set $A$. (All sets $A_i$ have $30$ elements).
Now, I am looking for a set $C=\{ A_1,A_2,A_3,...,A_n \}$ so, that if I pick randomly $6$ elements of set $A$, they will be (at least) in one of sets $A_i$.
What is $n$? Let's see: first of all, how much possibilities are to pick $6$ elements of set $A$? There are $\binom{49}{6}=\large\frac{49 \cdot 48 \cdot 47 \cdot 46 \cdot 45 \cdot 44}{6!}=13,983,816$
Secondly, how much of these possibilities covers one of sets $A_i$? Because set $A_i$ has $30$ elements, it covers  $\binom{30}{6}=\large\frac{30 \cdot 29 \cdot 28 \cdot 27 \cdot 26 \cdot 25}{6!}=593,775$
Now dividing both results, it gives $23.55$ and this means, that we need at least $n=24$ (probably more, I am not sure).
So the question is, how do you find the set $C$?
Let's say, we can start so: $A_1=\{1,2,3,...,30\}$, this will be first set. But what next? With some algorithm I can implement it in C or Java, but I don't know how to start. Thanks.

REPLY [7 votes]: One possible approach is to partition $A$ into 10 sets (say $B_i$), each of size 5 (one will be of size 4). Define your $A_j$ as union of any 6 of the above $B_i$. So you get $\binom{10}{6}$ such $A_j$ (some of them will have size 29, but you can throw in some random element if you want to make it 30). Any 6 element set touches at most 6 blocks $B_i$ and hence is present in one of the above $A_j$. This gives an upper bound for size of $C$ as $\binom{10}{6} = 210$.