SEAMO 2027 - Paper F - SET 07 - Self-study Question 8

In a shooting tournament, eight clay targets hang in three columns: two columns have three targets each, and one has two. To break the targets, first choose a column and then break its lowest remaining unbroken target. If these rules are followed, in how many different orders can all eight targets be broken?

Original source figure (schematic), preserving its mathematical labels and arrangement.