[Math] How to divide 30 people into 6 different groups of 5 people in 6 ways so that no two groups share two people

combinatoricsdiscrete mathematics

I have a group of 30 people that I need to divide into 6 groups of 5 people in 6 different ways, however I do not want the same people to be together twice.

I already have 5 ways written down, but I just need one more.

Added later by Rebecca: Here's a couple of randomly generated examples for 5 different ways:

[[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20],[21,22,23,24,25],[26,27,28,29,30]]
[[1,6,11,16,21],[2,7,12,17,27],[3,8,13,23,28],[4,14,19,24,29],[5,9,18,22,26],[10,15,20,25,30]]
[[1,7,13,19,25],[2,8,14,20,26],[3,9,15,21,27],[4,10,16,22,28],[5,11,17,23,29],[6,12,18,24,30]]
[[1,8,15,17,24],[2,9,16,23,30],[3,10,12,19,26],[4,11,18,25,27],[5,7,14,21,28],[6,13,20,22,29]]
[[1,9,12,20,28],[2,10,18,21,29],[3,6,14,17,25],[4,7,15,23,26],[5,13,16,24,27],[8,11,19,22,30]]

[[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20],[21,22,23,24,25],[26,27,28,29,30]]
[[1,6,14,20,29],[2,7,13,19,21],[3,15,17,25,28],[4,10,18,22,26],[5,8,11,24,30],[9,12,16,23,27]]
[[1,7,15,16,22],[2,8,17,23,29],[3,10,14,21,30],[4,9,11,20,28],[5,13,18,25,27],[6,12,19,24,26]]
[[1,9,13,17,30],[2,10,16,24,28],[3,6,11,18,23],[4,7,12,25,29],[5,15,20,21,26],[8,14,19,22,27]]
[[1,10,11,19,25],[2,12,20,22,30],[3,8,13,16,26],[4,6,17,21,27],[5,7,14,23,28],[9,15,18,24,29]]

Best Answer

Is this what you are after? I am not sure so I will leave out the explanation and references until I am.

[[1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [11, 12, 13, 14, 15], [16, 17, 18, 19, 20], [21, 22, 23, 24, 25], [26, 27, 28, 29, 30]]

[[1, 6, 11, 16, 21], [2, 7, 13, 19, 30], [3, 8, 14, 24, 29], [4, 9, 20, 25, 28], [5, 15, 18, 23, 27], [10, 12, 17, 22, 26]]

[[1, 8, 13, 20, 22], [2, 6, 12, 23, 28], [3, 9, 11, 18, 30], [4, 15, 16, 24, 26], [5, 7, 17, 25, 29], [10, 14, 19, 21, 27]]

[[1, 7, 18, 24, 28], [2, 10, 11, 20, 29], [3, 6, 15, 19, 22], [4, 14, 17, 23, 30], [5, 9, 13, 21, 26], [8, 12, 16, 25, 27]]

[[1, 10, 15, 25, 30], [2, 9, 17, 24, 27], [3, 7, 12, 20, 21], [4, 6, 13, 18, 29], [5, 14, 16, 22, 28], [8, 11, 19, 23, 26]]

[[1, 9, 12, 19, 29], [5, 6, 20, 24, 30], [4, 7, 11, 22, 27], [8, 15, 17, 21, 28], [3, 10, 13, 16, 23], [2, 14, 18, 25, 26]]

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