Calculate permutations where order matters. Grade 12 Math 30-1.
Lesson 7.2 of Probability Distributions in Math 30-1 — Alberta curriculum lessons.
Definition. A permutation is an arrangement of all or part of a set of objects where order matters. Swapping the positions of two items creates a different permutation.
The Formula. If n objects are arranged r at a time, the number of permutations is:
_nP_r = (n!)/((n-r)!)
Key Strategies. When a problem has restrictions, always ask yourself two questions before starting:
1. Are there any positions with special rules? → Fill those first.
2. Are we allowed to reuse items? → Repetition allowed vs. no repetition.
Golden Rule: Always fill restricted positions before unrestricted ones.
Repetition vs. No Repetition:
• Repetition allowed: An item can be used more than once. Each position has the same number of choices.
• No repetition: Once an item is placed, it cannot be reused. The number of choices decreases by 1 with each position filled.
Formulas. Permutations of parts of a group
The number of permutations of n different objects taken k at a time is:
(n!)/((n-k)!)
Permutations with repetition
The number of ways to arrange n objects, where a of them are identical of type 1, b of them are identical of type 2, c of them are identical of type 3, … is:
(n!)/(a! b! c!)
Example 1 — Evaluate _8P_3. Use the formula to evaluate _8P_3.
Step 1 — Identify n and r.
n = 8 (total objects), r = 3 (arranging 3 at a time)
Step 2 — Substitute into the formula.
_8P_3 = (8!)/((8-3)!) = (8!)/(5!)
Step 3 — Expand and cancel.
= (8 × 7 × 6 × 5!)/(5!) = 8 × 7 × 6 = 336
→ 336
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