Explain the difference between permutation and configuration. Calculate the number of permutations and configurations possible while selecting three days out of seven days in a week.

Understanding Permutation and Configuration

1. Definition of Permutation

Permutation refers to the arrangement of a subset of items where the order of arrangement is important. It calculates the number of ways to arrange a certain number of objects from a larger set, considering the sequence in which they appear. For instance, arranging the letters A, B, and C would yield different results such as ABC, ACB, BAC, BCA, CAB, and CBA — each arrangement being unique due to the order of the letters.

2. Definition of Configuration (Combination)

Configuration, often referred to as combination, involves selecting items from a larger set where the order of selection does not matter. It’s used to calculate the number of ways to choose a subset where the sequence is irrelevant. For example, selecting 2 letters from A, B, and C would result in combinations like AB, AC, and BC, where AB is considered the same as BA.

Differences Between Permutation and Configuration

• Order Importance: In permutations, the order of items is important, while in configurations, the order is not considered.
• Formulae: The formula for permutations of \(r\) items from \(n\) is \(P(n, r) = \frac{n!}{(n-r)!}\), and the formula for configurations of \(r\) items from \(n\) is \(C(n, r) = \frac{n!}{r!(n-r)!}\).

Calculation for Selecting Three Days Out of Seven

1. Permutation Calculation

To find the number of permutations of selecting 3 days out of 7 (considering the order in which they are selected matters), we use the permutation formula:

\[
P(7, 3) = \frac{7!}{(7-3)!} = \frac{7 \times 6 \times 5 \times 4!}{4!} = 7 \times 6 \times 5 = 210
\]

There are 210 different ways to select and arrange 3 days from a week.

2. Configuration Calculation

To find the number of configurations (combinations) of selecting 3 days out of 7 (where the order does not matter), we use the combination formula:

\[
C(7, 3) = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

There are 35 ways to select 3 days from a week without considering the order of the days.

Conclusion

The choice between permutation and configuration depends on whether the order of items is important in a given context. Permutations are used when the sequence affects the outcome, while configurations are appropriate for scenarios where only the choice of items matters, not the sequence. In the example of selecting days from a week, the substantial difference in results between permutations and configurations (210 vs. 35) highlights the impact of considering order.