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In probability and combinatorics, the combination formula is used to determine the number of ways to select a specific number of items from a larger set. The formula is represented as \binom{n}{k}\, where \ is the total number of items, and \k\ is the number of items to select.
For example, if you have 7 people and want to select 2, we write it as \binom{7}{2}\. The formula itself is: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Here, \!\ (n factorial) means multiplying all whole numbers from \ down to 1.
In the given exercise, to find the number of ways to select 2 people out of 7, we used the combination formula: \[ \binom{7}{2} = \frac{7!}{2! \/ (7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \] You can see how factorials simplify and cancel out in the division process. This is how we determine there are 21 ways to select 2 people from 7.

Once we know the total number of ways to select 2 people, we can calculate the probability of selecting 2 specific people, such as 2 women. Probability is essentially a fraction with:

• The number of favorable outcomes on top
• The total number of possible outcomes on the bottom

To find the probability that both people selected are women, we first calculate \binom{4}{2}\, the number of ways to select 2 women out of 4. This results in: \[ \binom{4}{2} = \frac{4!}{2! \/ (4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \] Now, we divide the number of favorable outcomes by the total possible outcomes: \[ P(\text{Both are women}) = \frac{ \binom{4}{2} }{ \binom{7}{2} } = \frac{6}{21} = \frac{2}{7} \] So, the probability is \frac{2}{7}\, meaning there's roughly a 28.57% chance of selecting 2 women.

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. It heavily uses the concept of combinations and permutations.
In this exercise, we use combinatorics to find different ways to select people from a group. Applying the combination formula helps solve real-world problems in fields like statistics, computer science, and operations research.
The exercise shows how combinatorics answers questions about arranging a committee, selecting team members, or any scenario needing calculated choices. It simplifies complex counting processes using established formulas and principles. By understanding the basics of combinatorics, students can tackle various selection, arrangement, and probability problems effortlessly.