Question

A plant is to contain 5 identical distillation columns each of which can be placed in any of 8 locations. How many possible arrangements of the 5 columns among the 8 locations are there?

Answer 1

Solution:

Since the 5 distillation columns are identical, it is immaterial which column is placed in which of the 8 locations. As a result, the positions of the columns in the locations do not matter.

It is also mentioned that each of the 5 identical distillation columns “can be placed in any of 8 locations”. Now, based on the interpretation of this condition, there may be two situations:

Situation I:

Each of the 8 locations can hold 0, 1, 2, 3, 4, or 5 distillation columns.

This means that the 1^st distillation column can be placed in any of the 8 locations, the 2^nd distillation column can be placed in any of the 8 locations, the 3^rd distillation column can be placed in any of the 8 locations, the 4^th distillation column can be placed in any of the 8 locations, and the 5^th distillation column can be placed in any of the 8 locations.

Thus, if each location can hold multiple distillation columns, the number of possible arrangements of the 5 distillation columns among the 8 locations is 32,768 (= 8⁵).

Situation II:

Each of the 8 locations can hold a maximum of 1 distillation column.

Since the columns are identical, this scenario is the same as selecting any 5 out of the 8 locations, and placing one distillation column in each selected location. The number of ways in which this selection can be made is (₈C₅) = 56.

Thus, if each location can hold a maximum of 1 distillation column, the number of possible arrangements of the 5 distillation columns among the 8 locations is 56.