What is the area of mathematics that deals with counting and arrangement of objects?

The area of mathematics that deals specifically with counting, arranging, and selecting objects is combinatorics. This branch of mathematics focuses on the various ways in which objects can be combined or arranged, using principles and formulas to determine the total number of combinations or permutations possible under certain conditions. For example, combinatorics is used to solve problems related to counting the number of ways to arrange a group of people, the number of possible combinations of a set of items, or the distribution of objects into different groups.

While probability involves the likelihood of events occurring and statistics deals with data analysis and interpretation, neither focuses primarily on the arrangement and counting of objects as combinatorics does. Additionally, calculus encompasses analysis of change and motion, dealing with concepts such as differentiation and integration, which do not directly relate to counting and arrangement. Thus, combinatorics is the correct choice, as it is fundamentally concerned with the counting and arrangement of objects.