91Ó°ÊÓ

In combinatorics, permutations refer to the arrangement of a set of items where the order is important. To compute permutations, we use the formula \(\text{Permutations} = P(n, r) = \frac{n!}{(n - r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to arrange. For example, in the Digital Pet Rock Company scenario, there are 10 candidates for four officer positions. Since the order of these positions (President, CEO, COO, CFO) matters, we calculate the permutations as follows: \[ P(10, 4) = \frac{10!}{(10-4)!} = \frac{10!}{6!} = 10 \times 9 \times 8 \times 7 = 5040 . \] This calculation shows there are 5,040 unique ways to appoint the four officers, which highlights the importance of considering order in permutations.

Unlike permutations, combinations do not consider the order of items. The formula for combinations is \(\text{Combinations} = C(n, r) = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to select. In the example of forming a committee of four members from 10 candidates, the order does not matter. Using the combinations formula: \[ C(10, 4) = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = 210 . \] This calculation shows there are 210 different ways to appoint the committee of four. Combinations allow us to focus on the selection of items, regardless of the order in which they were picked.

Probability measures the likelihood of a particular event occurring and is given by the ratio of favorable outcomes to the total number of possible outcomes. In our example, to find the probability of randomly selecting the four youngest candidates for the committee, we first identify that there is only one favorable outcome: selecting these four specific candidates. There are 210 possible ways to form a committee of four from ten candidates. Thus, the probability is calculated as: \[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{210} . \] So, the probability of selecting the four youngest candidates is \(\frac{1}{210}\). This illustrates how probability can be determined using both permutations and combinations to understand the range of possible outcomes.