Grades 7–8 Skill Alignment: Enrichment (extension of CCSS 7.SP.C.8)
Arrangements & Factorials — How Many Orders? (Grades 7–8)
When order matters — like lining people up — count the arrangements (permutations).
What it is
Understanding arrangements & factorials
For the first spot there are n choices, then n−1 for the next, and so on, which is n! ("n factorial"). Three books arrange in 3! = 3 × 2 × 1 = 6 ways; four in 4! = 24.
Worked Example
Seeing it in action
1
Worked example
How many ways to line up 3 books?
3! = 3 × 2 × 1 = 6.
Visual model
When order matters, multiply the descending choices.
Watch for this
A common mistake, and the fix
Using the same number of choices for every place in the line.
For 3 books, 3 × 3 × 3 = 27 counts lines that use a book twice. Once a book is placed it can’t be used again, so the choices shrink: 3 × 2 × 1 = 6.
Interactive Check
Try a few
Arrange 4 people in a row
Answer: 24
4!.
Arrange 2 items
Answer: 2
Arrange 5 books
Answer: 120
5!.
Try it in Numeris.
A short lesson, then practice with feedback on every answer.
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