Set and Combination problem

[sanjibannath]

I have a question. Suppose I have 3 sets of sets namely A={{ab} {bc} {ac}} B={{defg} {defh} {degh} {dfgh} {efgh}} and C={{ijklmn} {ijklmo} {ijklno} {ijkmno} {ijlmno} {iklmno} {jklmno}}. For A: 3 Choose 2 (3 sets of 2 letters) for B: 5 Choose 4 (5 sets of 4 letters) and for C: 7 Choose 6 (7 sets of 6 letters). The total number of multiple choices are 105 (3x5x7). Out of 105 a few examples are:

{{ab},{defg},{ijklmn}}

{{ab},{defg},{ijklmo}}

{{ab},{defg},{ijklno}}
…….
{{bc},{defh},{ijklmn}}

{bc},{defh},{ijklmo}}

{{bc},{defh},{ijklno}}

….. and so on.

Now my question is what is the minimum number of letters do I need to pick from every 3 sets of 105 choices so that those newly chosen sets of letters will represent all of 105 choices? I mean when I pick the letter ‘a’ it will represent sets {ab} and {ac} of the 105 choices and similarly when I choose ‘b’ it will represent sets {ab} and {bc} of the 105 choices. Likewise, when I pick the letter ‘d’ it will represent all the 4 sets where ‘d’ is there: {defg} {defh} {degh} and {dfgh} of the 105 choices. I have found that I have to choose a minimum of 5 sets of 3 letters from each set of sets to represent all of 105 multiple choices. And they are {{a},{d},{i}}, {{b},{e},{j}}, {{c},{f},{k}}, {{a},{g},{l}} and {{b},{h},{m}}. So, is there any formula to find it out? I have found it by computing and striking out one choice after another without any formula. Thanks in advance.

Regards