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To solve problems involving selecting a subset from a larger set, we often use the combination formula. Unlike permutations, where the order matters, combinations focus on selecting items where the order does not matter. This is why the combination formula is the tool of choice for problems like selecting team members from a larger group.

The combination formula is written as follows:

• \( \binom{n}{k} = \frac{n!}{k!\,(n-k)!} \)

Here:

• \( n \) is the total number of items to select from,
• \( k \) is the number of items to select,
• \(!\) represents the factorial operation, a sequence of multiplying numbers down to 1.

This formula allows us to calculate the number of ways to choose \( k \) elements from a set of \( n \) elements without considering the order. For example, choosing 3 members from a team of 8 is done by computing \( \binom{8}{3} = 56 \).

The binomial coefficient is a fundamental aspect of combinatorics and directly ties into the concept of combinations. It's denoted as \( \binom{n}{k} \) and is often interchangeable with the term 'combination'.

Why is it called a binomial coefficient? Because it appears in the binomial theorem, which expands expressions of the form \((a+b)^n\). In this context, the coefficient represents how many ways you can choose \( k \) elements from \( n \) total, just like in the combination formula.

• This is why calculating \( \binom{10}{3} \) when selecting members from the second team is straightforward once you understand the binomial coefficient. It was calculated as 120.

Whether it's used in expanding binomial expressions or selecting team members, the binomial coefficient is all about counting selections where order does not matter.

In many combinatorial problems like forming matches from teams, decisions can be considered independent choices. But what does this mean?

In the context of our exercise, independent choices mean that the way we select members from the first team does not affect how we choose from the second team. When the selection of one group is not influenced by another, these choices are referred to as independent.

To find the total number of ways to arrange something when dealing with independent choices, you multiply the number of ways to choose each group.

• In the given problem, choosing members from two separate teams independently gives us the result by multiplying the two combinations: \( 56 \times 120 = 6720 \).

Each combination is independent of the other, allowing us to simply multiply to reach the final result. This principle of independence is widespread in probability and combinatorics alike.