Example Card: Finding the Number of Permutations
Calculate the number of permutations in Grade 9 math. Apply the formula nPr = n!/(n-r)! to count ordered arrangements where order matters in probability and counting problems.
Key Concepts
When the order of finishers matters, the number of possibilities changes. Let's calculate it using the core idea of permutations.
Example Problem In a race with 10 runners, how many different ways can the first, second, and third place medals be awarded?
Step by Step 1. This is a permutation because the order in which the runners finish is important. We need to find the number of permutations of 10 runners taken 3 at a time. 2. Write the permutation formula: $ nP r = \frac{n!}{(n r)!}$. 3. Substitute $n=10$ and $r=3$ into the formula to get the simplified factorial expression. $$ {10}P 3 = \frac{10!}{(10 3)!} = \frac{10!}{7!} $$ 4. Now, write out the factors for each factorial. Then, cancel the common terms to simplify. $$ \frac{10 \cdot 9 \cdot 8 \cdot \cancel{7} \cdot \cancel{6} \cdot \cancel{5} \cdot \cancel{4} \cdot \cancel{3} \cdot \cancel{2} \cdot \cancel{1}}{\cancel{7} \cdot \cancel{6} \cdot \cancel{5} \cdot \cancel{4} \cdot \cancel{3} \cdot \cancel{2} \cdot \cancel{1}} = 10 \cdot 9 \cdot 8 $$ 5. Multiply the remaining numbers to find the final count. $$ 10 \cdot 9 \cdot 8 = 720 $$ There are 720 different ways to award the top three medals.
Common Questions
What is the permutation formula and how do you use it?
nPr = n!/(n-r)! where n is total items and r is number being arranged. Order matters in permutations — arranging AB is different from BA.
What is the difference between permutations and combinations?
Permutations count ordered arrangements (AB ≠ BA), combinations count unordered selections (AB = BA). Use permutations when order matters, like ranking contestants.
How do you calculate 8P3?
8P3 = 8!/(8-3)! = 8!/5! = 8 × 7 × 6 = 336. Multiply the top 3 descending values from 8 because you are filling 3 ordered positions from 8 items.