Permutation
A permutation is an arrangement of objects in a specific order, where the order matters. The number of permutations of n objects taken r at a time is calculated using the formula P(n, r) = n! divided by (n - r)!. For example, the number of ways to arrange 3 different books from a set of 5 is P(5, 3) = 5 x 4 x 3 = 60. This Grade 7 math skill from Saxon Math, Course 2 extends the Fundamental Counting Principle and introduces combinatorics, with direct applications in scheduling, security codes, competition rankings, and probability calculations.
Key Concepts
Property A permutation is one of the possible ways to order a set of items, like digits or letters. In a permutation, the order always matters.
Examples The digits 2, 3, and 5 have six permutations: $235, 253, 325, 352, 523, 532$. The letters C, A, T have six permutations, including CAT, ACT, and TAC.
Explanation Think of it as shuffling! Each different arrangement is a new permutation. Making an organized list from smallest to largest is a great strategy to find every single combination without missing any.
Common Questions
What is a permutation?
A permutation is an arrangement of objects where the order matters. Choosing president, vice president, and secretary from 10 candidates is a permutation because the order (who gets which role) matters.
How do I calculate the number of permutations?
For ordered arrangements without repetition, multiply the number of choices for each position. For 3 positions from 5 objects: 5 x 4 x 3 = 60 permutations.
What is the difference between a permutation and a combination?
In a permutation, order matters (ABC and BAC are different). In a combination, order does not matter (ABC and BAC are the same group). Use permutations for rankings, combinations for selections.
When does order matter in counting problems?
Order matters in problems involving rankings, sequences, codes, and assignments of different roles. If changing the order gives a different outcome, use permutations.
When do students learn about permutations?
Permutations are introduced in Grade 7 as an extension of the Fundamental Counting Principle. Saxon Math, Course 2 covers them in Chapter 10 alongside combinations.
How does the Fundamental Counting Principle relate to permutations?
Permutations use the Fundamental Counting Principle applied to ordered selections: multiply the number of options for each sequential choice, decreasing by one each time for selections without replacement.
What are real-world examples of permutations?
The number of possible 4-digit PIN codes (0-9 with no repeats), the number of ways to arrange a top-5 finish in a race, or seating arrangements for a specific number of people at a table are all permutations.