Final answer:
Combinations are used to count the number of ways to form committees from a given number of people since the order of selection doesn't matter, which is essential in calculating the probability of various committee compositions.
Step-by-step explanation:
Combinations are used to count the number of ways to form a committee from a given number of people because order does not matter in such groups. When selecting members for a committee, it is only important who is on the committee, not the sequence in which they were chosen. This differentiates combinations from permutations, where the order of selection is important.
For instance, when forming a committee of seven students from a group that consists of 18 women and 15 men, we can use combinations to calculate the total number of possible committees. If we want to calculate the probability that the committee has more than four men, we use combinations to find the total number of ways we can choose men and women to fill the seven spots, and then find the probability of the specific cases where there are more than four men.
To demonstrate how combinations are used in probability calculations, consider the scenario where we want two students to become a chairperson and recorder from a committee composed of 10 staff members and six students. The selection process is carried out by drawing names twice, and since these events are connected (the first draw affects the second), we cannot simply multiply individual probabilities. Instead, we calculate the probability that both draws result in students, and combinations help determine the number of candidate arrangements for this outcome.