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When discussing the range of integers in mathematics, we are referring to a sequence of whole numbers between two specified boundaries. The boundaries in the problem are from 5 to 200. This means the smallest number in the sequence is 5, and the largest is 200. In a range of integers, both the starting and ending numbers are typically included. Thus, we describe this as a closed interval. The notation for this is

• \([5, 200]\), which signifies all numbers from 5 to 200, inclusive.

Understanding the range is crucial. It allows us to enumerate all possibilities within those defined limits. Such clarity helps when solving problems that require counting or listing of numbers. With this comprehension, we can confidently tackle any range-related queries by knowing exactly what numbers are involved.

The threshold in counting refers to a specific value that sets a limit or boundary beyond which only certain values are considered. In our exercise, the threshold is 72. This means that when we are asked to determine which numbers are greater than 72, we exclude 72 itself. Instead, we start counting from the next integer. This is common in problems where we need to filter out certain values, allowing us to focus only on the relevant numbers above the threshold.

• Identifying the threshold clarifies which numbers fall into the "counted" or "not counted" categories.
• In most scenarios, this involves using inequalities such as "greater than" (>).

When applying this to our range of integers, we effectively narrow down the sequence of numbers that we need to consider, simplifying the counting process.

Inclusive counting is a method used to ensure that both endpoints of a certain set or range of numbers are counted. When the problem states to find how many numbers are greater than a certain value, but the numbers within the identified range should also be included, we apply inclusive counting.
In our example, once the threshold of 72 is identified, inclusive counting starts from the next number, 73, up to 200.

• The endpoint, 200, must also be counted.

The basic formula for inclusive counting in a range from a number, say \(a\), to \(b\) is:

• \((b - a + 1)\).

In this case, the application becomes \(200 - 73 + 1 = 128\), ensuring that each number in the list, from 73 to 200, is included in the count. Using inclusive counting ensures accuracy and completeness when determining the numbers within any defined mathematical boundaries.