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When you need to select a specific number of items from a larger group, the concept of combination comes in handy. The **combination formula** allows you to calculate how many different ways you can pick items without considering the order. This is why combinations are often phrased as "choosing" or "selecting."A combination is mathematically expressed with the formula:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]Where:

• \( n \) is the total number of items to choose from,
• \( r \) is the number of items to be selected, and
• \( ! \) denotes factorial, which is the product of an integer and all the integers below it.

In the exercise, to form a committee of 5 people from a group of 375, you would use the formula:\[\binom{375}{5} = \frac{375 \times 374 \times 373 \times 372 \times 371}{5 \times 4 \times 3 \times 2 \times 1}\]This calculation helps determine the number of possible combinations for selecting our committee. Remember, the order of selection doesn't matter in combinations, which makes them quite different from permutations.

**Disjoint sets** are groups of items where no elements overlap between them. So, if you have two sets that are disjoint, no item from the first set is also in the second set.In the exercise, if committees must be disjoint, once you choose your first committee of 5 members, those members are removed from the pool.This means when you're selecting the next committee, you select from the remaining members only. That is, the second committee is chosen from a new pool of 370 people because 5 have already been selected for the first.To determine the total number of ways to create two disjoint committees from 375 candidates:

• First, choose the first committee: \( \binom{375}{5} \)
• Then choose the second committee from the remaining 370 people: \( \binom{370}{5} \)

Multiply these two results to find the total number of disjoint committee selections:\[\binom{375}{5} \times \binom{370}{5}\]This results in choices that ensure those committees have absolutely no overlapping members.

Unlike disjoint sets, **overlapping sets** can share some or all elements between them. In the context of the exercise, if committees are allowed to overlap, members can be part of both committees.To create committees with overlapping members:

• First, select 10 individuals from the entire group of 375: \( \binom{375}{10} \)
• Then, from these 10, you can form two committees of 5 members each. The number of ways to choose one committee out of 10 chosen people is: \( \binom{10}{5} \)

Thus, the overall number of ways to form overlapping committees is calculated by:\[\binom{375}{10} \times \binom{10}{5}\]This method allows flexibility, letting members participate in both committees, if desired. It captures all potential combinations where committee memberships can intersect, offering a broader range of selection possibilities.