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Combinatorics is a branch of mathematics that deals with counting, arranging, and combining objects. It is useful in determining the number of possible outcomes in various scenarios. In problems involving probability, combinatorics helps us figure out how many ways something can happen.
In the given problem, we are determining how many ways we can choose 4 shoes out of 10. This involves calculating combinations and permutations based on the problem's constraints.
Understanding combinatorics allows us to break down a complex problem into manageable parts. We can then find the total possible outcomes and the number of favorable outcomes.

The binomial coefficient is a key element in combinatorics that helps us calculate the number of ways to choose a certain number of objects from a larger set, without regard to the order of selection. It is denoted as \(\binom{n}{k}\), where \(n\) is the total number of objects and \(k\) is the number of objects to choose.
The formula for the binomial coefficient is:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
In our problem, the total number of ways to choose 4 shoes from 10 is calculated using \(\binom{10}{4}\). This tells us how many different groups of 4 shoes we can form from the 10 available shoes.
Next, we use another binomial coefficient \(\binom{5}{4}\) to calculate the number of ways to choose 4 pairs out of 5. Once we have our 4 pairs, we choose one shoe from each, giving us the total favorable outcomes when no pair is chosen.

The complement rule is an essential concept in probability theory. It is used to find the probability of the complement of an event. The complement of an event \(A\) is basically everything that is not \(A\). The probability of the complement of \(A\) can be calculated by subtracting the probability of \(A\) from 1:
\[ P(A^c) = 1 - P(A) \]
In our problem, instead of directly calculating the probability that at least one pair of shoes is chosen, we first calculate the probability that no pairs are chosen. This is P(no pair). Then, we use the complement rule to find the probability of the event we are interested in (at least one pair being chosen).
Using the complement rule simplifies the problem and often makes calculations easier. By focusing on the simpler scenario first (no pairs chosen), we can more easily determine the probability of the more complex scenario (at least one pair chosen).