Permutations
Count permutations using nPr = n!/(n-r)!: calculate ordered arrangements of r items from n distinct objects, distinguishing permutations from combinations in probability problems.
Key Concepts
A permutation is an arrangement of objects where the order is important. The number of permutations of $n$ objects taken $r$ at a time is: $$P(n, r) = \frac{n!}{(n r)!}$$ For permutations with repeated items, the formula is $\frac{n!}{q 1! \cdot q 2! \cdots q k!}$.
How many ways can a President and VP be chosen from 8 candidates? $P(8, 2) = \frac{8!}{(8 2)!} = \frac{8!}{6!} = 56$. Find the distinguishable permutations of the letters in 'CHEESE': $\frac{6!}{3!} = \frac{720}{6} = 120$. How many 3 digit lock codes can be made from digits 1 7 if no digit repeats? $P(7, 3) = \frac{7!}{(7 3)!} = 210$.
Permutations are all about position! Think of it like a race: getting 1st, 2nd, and 3rd place is a different outcome than the same three people in a different order. Use this when arranging things where the sequence is critical, like letters in a password or runners finishing a race. Order is everything here!
Common Questions
What is a permutation and how does it differ from a combination?
A permutation counts ordered arrangements where the sequence matters. A combination counts unordered selections where only which items are chosen matters. Arranging 3 books on a shelf is a permutation; choosing 3 books to read is a combination.
How do you calculate nPr?
The permutation formula is nPr = n! divided by (n-r)! where n is the total number of distinct items and r is the number being selected and arranged. For 5P3: 5! divided by (5-3)! = 120 divided by 2 = 60 ordered arrangements.
When does the order of selection matter in a real-world problem?
Order matters for rankings such as 1st, 2nd, 3rd place; passwords where digit sequence is significant; race finishing positions; and seating arrangements. Any time rearranging the selected items creates a meaningfully different outcome, use permutations.