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Sample selection plays a crucial role in combinatorics, especially in situations where we want to choose a subset of items from a larger group. In our exercise, we're selecting 9 items from a total of 20. But what's important is that we select them without replacement, which means once an item is chosen, it can't be picked again.

The concept of sample selection is commonly used when order doesn’t matter. This is known as a combination, rather than a permutation, where order would matter. When tackling these types of problems, it's key to clearly define the total number of items available and the number of items to be selected. It helps simplify and solve the problem using the right mathematical tools.

In practice, sample selection occurs in various scenarios like forming committees, picking lottery numbers, or even dealing cards from a deck.

Factorial calculation is one of the cornerstones in solving combinatorics problems. A factorial, denoted by the exclamation mark '!', refers to the product of all positive integers up to a specified number. For example, 5! is calculated as 5 × 4 × 3 × 2 × 1 = 120.

In our exercise, to find how many ways we can select 9 items from 20, we need to calculate factorials for 20! (20 factorial), 9! (9 factorial), and 11! (since 20 - 9 = 11).

Calculating factorials can be done using calculators, especially for larger numbers, because numbers grow extremely quickly. Understanding how factorials work is essential in breaking down combination problems into manageable pieces. Remember:

• The factorial of a number n is expressed as n!
• Factorial of zero (0!) is always 1
• Factorials are only defined for non-negative integers

The combination formula is used to determine the number of ways to select a group of items from a larger set when order does not matter. The formula is written as:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

Where:

• \( n \) is the total number of items
• \( k \) is the number of items to be selected

For our problem, this translates to \( C(20, 9) = \frac{20!}{9! \, 11!} \).

This formula calculates not just any selection, but specifically those where the order of picking doesn't affect the outcome. That's why it's different from permutations, where order matters.

By plugging our values into the combination formula, we can methodically determine the number of ways to select 9 items from a set of 20. This systematic approach provides clarity and precision, essential for solving real-world problems efficiently.