1 A set consists of five distinct non-integer values, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E . The set is partitioned into non-empty subsets and there are at least two subsets in each partition.

1. Show that there are 15 different partitions into two subsets.
2. Show that there are 25 different partitions into three subsets.
3. Calculate the total number of different partitions. The numbers 12, 24, 36, 48, 60, 72, 84 and 96 are marked on a number line. The number line is then cut into pieces by making cuts at \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , where \(0 < \mathrm { A } < \mathrm { B } < \mathrm { C } < \mathrm { D } < \mathrm { E } < 100\).
4. Explain why there must be at least one piece with two or more of the numbers 12, 24, 36, 48, 60, 72, 84 and 96.