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Probability theory deals with the chance of a certain event occurring. In simple terms, it tells us how likely or unlikely something is to happen.

When we talk about probability, we're talking about a number between 0 and 1. This number represents the likelihood of an outcome, where 0 means it's impossible and 1 indicates certainty. To find the probability of an event, you divide the number of favorable outcomes by the total number of all possible outcomes.

In our exercise, we are interested in two events related to forming a committee from a group of people:

• The probability of including both married couples.
• The probability of including the youngest members.

To calculate these probabilities, we first have to determine the number of favorable outcomes (how many times these couples or youngest members can be chosen), and then divide that by the total ways the committee can be formed (which was calculated as 5005 in this case). This will give us a sense of how likely these specific groups will be part of the committee.

The binomial coefficient is a key concept in combinatorics, which allows us to calculate the number of ways to choose a subset of items from a larger set. It is often referred to as "n choose r" and is written as \( \binom{n}{r} \).

In the exercise, we use the binomial coefficient to determine how many ways we can select committee members.

The formula for the binomial coefficient is:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n! \) (n factorial) is the product of all integers up to \( n \). This formula gives us the number of ways to choose \( r \) elements from a set of \( n \) elements.

In our specific problem:

• \( \binom{15}{6} \) calculates the total number of ways to form a committee of six from fifteen people.
• \( \binom{11}{2} \) computes the number of ways to choose two additional members after selecting the two married couples out of the remaining eleven.
• \( \binom{12}{3} \) calculates the ways to choose additional committee members after selecting the three youngest ones.

Using the binomial coefficient simplifies the process of counting combinations, avoiding tedious manual enumeration.

Mathematical problem-solving involves a step-by-step approach to finding solutions to problems. This structured method helps in breaking down complex problems into manageable parts, ensuring that no detail is overlooked.

For the exercise at hand, we carefully followed these steps:

• First, we identified the total number of possible committees using the binomial coefficient \( \binom{15}{6} \).
• Next, we calculated the number of favorable outcomes for each event (having both married couples and the youngest members). This involved using the binomial coefficient to find how many ways these particular groups can be selected.
• Finally, we determined the probability by dividing the favorable outcomes by the total possible combinations, allowing us to quantify how likely each scenario is.

This systematic approach illustrates the power of problem-solving skills in mathematics. By using logical reasoning and mathematical tools like binomial coefficients, we can efficiently solve and understand a wide range of combinatorial problems.