Generate the fewest lottery tickets to play in order to have at least N good numbers

Question

This is a rather complex but very interesting maths subject (known as "covering problem"),

And I'd like your help for implementing it.

Imagine a lottery game, where each ticket must choose 5 random numbers in a set of 50 numbers (from 1 to 50).

It's quite easy to know the probability of a winning ticket, or the probability to have 1, 2, 3 or 4 good numbers.

It's also quite easy to "generate" all the tickets that have 1, 2, 3, 4 good numbers.

My question (and code challenge) is related to this, but slightly different:

I want to buy some lottery tickets (the fewest possible), such as at least one of my tickets has 3 good numbers.

Challenge

Your goal is to implement a generic solution (as a program or just a function), like this, in any language:

// Input: 3 prameters
min_lottery_tickets(total_numbers_to_choose_from, how_many_numbers_to_choose, how_many_good_numbers_i_want)

For the above example, one would just have to call:

min_lottery_tickets(50, 5, 3)

and the program will generate the smallest set of tickets to play to achieve this goal.

---

Example:

 min_lottery_tickets(10, 5, 2)

would output 7 tickets, like those:

1   2   3   4   5
5   6   7   8   9
10  1   2   6   7
10  3   4   8   9
3   4   6   7   8
1   2   3   8   9
1   4   9   5   10

because such tickets are enough to cover any pair of numbers from 1 to 10.

---

Output

Text, one line per ticket, tabulations or spaces between numbers

---

who wins

The most efficient program wins (i.e. the program generating the fewest tickets for the above parameters):

min_lottery_tickets(50, 5, 3)

Thanks!

Answer 1

I know it's not optimal, but here is the code in node.js:

function min_lottery_tickets(total_numbers_to_choose_from, how_many_numbers_to_choose, how_many_good_numbers_i_want) {
    c(function(result) {
        var other = result[result.length - 1];
        while (result.length < how_many_numbers_to_choose) {
            other++;
            var already = false;
            for (var i = 0; i < result.length; i++) {
                if (other === result[i]) {
                    already = true;
                    break;
                }
            }
            if (!already) {
                result.push(other);            
            }
        }
        if (other <= total_numbers_to_choose_from) {
            // Print results
            console.log(result);
        }
    }, total_numbers_to_choose_from, how_many_good_numbers_i_want);
}

function c(next, total_numbers, length, start, results) {
    if (!start) start = 1;
    if (!results) results = [];

    for (var i = start; i <= total_numbers + 1 - length; i++) {
        var resultsNew = results.slice(0);
        resultsNew.push(i);
        if (length > 1) {
            c(next, total_numbers, length - 1, i + 1, resultsNew);
        } else {
            next(resultsNew);
        }
    }
}

Some example results:

> min_lottery_tickets(5, 3, 2)
[ 1, 2, 3 ]
[ 1, 3, 4 ]
[ 1, 4, 5 ]
[ 2, 3, 4 ]
[ 2, 4, 5 ]
[ 3, 4, 5 ]

other:

> min_lottery_tickets(10, 5, 2)
[ 1, 2, 3, 4, 5 ]
[ 1, 3, 4, 5, 6 ]
[ 1, 4, 5, 6, 7 ]
[ 1, 5, 6, 7, 8 ]
[ 1, 6, 7, 8, 9 ]
[ 1, 7, 8, 9, 10 ]
[ 2, 3, 4, 5, 6 ]
[ 2, 4, 5, 6, 7 ]
[ 2, 5, 6, 7, 8 ]
[ 2, 6, 7, 8, 9 ]
[ 2, 7, 8, 9, 10 ]
[ 3, 4, 5, 6, 7 ]
[ 3, 5, 6, 7, 8 ]
[ 3, 6, 7, 8, 9 ]
[ 3, 7, 8, 9, 10 ]
[ 4, 5, 6, 7, 8 ]
[ 4, 6, 7, 8, 9 ]
[ 4, 7, 8, 9, 10 ]
[ 5, 6, 7, 8, 9 ]
[ 5, 7, 8, 9, 10 ]
[ 6, 7, 8, 9, 10 ]

other:

> min_lottery_tickets(10, 5, 3)
[ 1, 2, 3, 4, 5 ]
[ 1, 2, 4, 5, 6 ]
[ 1, 2, 5, 6, 7 ]
[ 1, 2, 6, 7, 8 ]
[ 1, 2, 7, 8, 9 ]
[ 1, 2, 8, 9, 10 ]
[ 1, 3, 4, 5, 6 ]
[ 1, 3, 5, 6, 7 ]
[ 1, 3, 6, 7, 8 ]
[ 1, 3, 7, 8, 9 ]
[ 1, 3, 8, 9, 10 ]
[ 1, 4, 5, 6, 7 ]
[ 1, 4, 6, 7, 8 ]
[ 1, 4, 7, 8, 9 ]
[ 1, 4, 8, 9, 10 ]
[ 1, 5, 6, 7, 8 ]
[ 1, 5, 7, 8, 9 ]
[ 1, 5, 8, 9, 10 ]
[ 1, 6, 7, 8, 9 ]
[ 1, 6, 8, 9, 10 ]
[ 1, 7, 8, 9, 10 ]
[ 2, 3, 4, 5, 6 ]
[ 2, 3, 5, 6, 7 ]
[ 2, 3, 6, 7, 8 ]
[ 2, 3, 7, 8, 9 ]
[ 2, 3, 8, 9, 10 ]
[ 2, 4, 5, 6, 7 ]
[ 2, 4, 6, 7, 8 ]
[ 2, 4, 7, 8, 9 ]
[ 2, 4, 8, 9, 10 ]
[ 2, 5, 6, 7, 8 ]
[ 2, 5, 7, 8, 9 ]
[ 2, 5, 8, 9, 10 ]
[ 2, 6, 7, 8, 9 ]
[ 2, 6, 8, 9, 10 ]
[ 2, 7, 8, 9, 10 ]
[ 3, 4, 5, 6, 7 ]
[ 3, 4, 6, 7, 8 ]
[ 3, 4, 7, 8, 9 ]
[ 3, 4, 8, 9, 10 ]
[ 3, 5, 6, 7, 8 ]
[ 3, 5, 7, 8, 9 ]
[ 3, 5, 8, 9, 10 ]
[ 3, 6, 7, 8, 9 ]
[ 3, 6, 8, 9, 10 ]
[ 3, 7, 8, 9, 10 ]
[ 4, 5, 6, 7, 8 ]
[ 4, 5, 7, 8, 9 ]
[ 4, 5, 8, 9, 10 ]
[ 4, 6, 7, 8, 9 ]
[ 4, 6, 8, 9, 10 ]
[ 4, 7, 8, 9, 10 ]
[ 5, 6, 7, 8, 9 ]
[ 5, 6, 8, 9, 10 ]
[ 5, 7, 8, 9, 10 ]
[ 6, 7, 8, 9, 10 ]