Permutation

Definition

A permutation refers to the arrangement of a set of objects in a specific order. The key aspect of permutations is that the order matters, meaning that different arrangements of the same set of objects are considered distinct. This concept is essential when calculating probabilities, as it helps to determine how many ways items can be arranged or selected, which is foundational for understanding both permutations and combinations.

5 Must Know Facts For Your Next Test

1. The formula for calculating permutations of n items taken r at a time is given by $$P(n,r) = \frac{n!}{(n-r)!}$$.
2. When all items are distinct, each arrangement creates a unique permutation, showcasing the importance of order.
3. Permutations are commonly used in scenarios such as seating arrangements, race outcomes, or any situation where the sequence is important.
4. The total number of permutations increases significantly with the addition of more items, illustrating the factorial growth.
5. When some items are identical, the formula adjusts to account for repetitions, given by $$P(n; n_1,n_2,...,n_k) = \frac{n!}{n_1!n_2!...n_k!}$$.

Review Questions

• How do permutations differ from combinations, and why is this distinction important in probability problems?
  • Permutations and combinations both deal with selections from a set, but the crucial difference lies in the importance of order. In permutations, the arrangement matters, meaning that different orders create distinct outcomes. This distinction is important in probability problems because it affects how we count possible outcomes; using permutations when order matters provides more accurate calculations for scenarios like seating arrangements or event scheduling.
• Given a set of 5 distinct books, how many different ways can they be arranged on a shelf? Explain the process to arrive at your answer.
  • To determine how many different ways 5 distinct books can be arranged on a shelf, we use the permutation formula for arranging all items, which is simply 5!. This calculates as $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$. Therefore, there are 120 unique arrangements for these books on the shelf.
• In what scenarios would you use a permutation formula that accounts for identical items, and how would this change your calculation?
  • You would use a permutation formula that accounts for identical items when dealing with sets that include repeated elements. For example, if you have 3 red balls and 2 blue balls, using the standard permutation formula would overcount the arrangements since swapping identical balls doesn't create a new arrangement. Instead, you would use $$P(n; n_1,n_2) = \frac{n!}{n_1!n_2!}$$ to adjust for these repetitions. This provides a more accurate count by dividing by the factorials of the counts of identical items.

Related terms

Factorial:

The product of all positive integers up to a given number, denoted as n!, which is used in calculating permutations.

Combination: A selection of items from a larger set where the order does not matter, contrasting with permutations where order is essential.

Arrangements: The different ways in which a set of items can be organized or ordered, which is directly related to the concept of permutations.