The idea, step by step
Factorial represents decreasing choices
For n different objects in a row, there are n choices for the first position, n−1 for the next, and so on. The product is n factorial, written n!. For example 4!=4×3×2×1=24. By convention 0!=1.
A shorter ordered selection uses only the needed factors. Choosing a captain and vice-captain from six students gives 6×5=30, not 6!, because only two roles are being filled.
Identical objects and unordered groups
Factorial counting assumes all objects are distinguishable and order matters. Swapping the two A letters in AAB does not make a new word. List the arrangements AAB, ABA, BAA to see why 3! would overcount.
For a simple two-person group, AB and BA name the same group. Count ordered selections first and divide by 2 because every pair was counted in two orders. Explain the overcount rather than applying a formula without checking its meaning.
A common trap
Using n! whenever n appears, even when only two roles are filled.
Instead: Count the number of positions actually being filled and whether their order matters.
See the thinking
Arrange different books
Five different books are placed in a row. How many orders?
- The successive choices are 5,4,3,2,1.
- Their product is 5!.
Roles versus a group
From 5 people, compare choosing a captain and deputy with choosing an unordered pair.
- Different roles give 5×4=20 ordered choices.
- For an unordered pair, each pair is counted twice.
Check your understanding
Evaluate 6!/4!.
Show answer
30.
Cancel 4!, leaving 6×5.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Evaluate 4!.2 marks
- Evaluate 7!/5!.2 marks
- Six different students line up. How many orders?2 marks
- From 6 students choose a captain and deputy. How many choices?2 marks
- How many unordered two-person groups can be chosen from 6 students?3 marks
- List the distinct arrangements of AAB and explain why there are not 6.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Evaluate 4!.2 marks
4×3×2×1=24.
24. - Evaluate 7!/5!.2 marks
Cancel 5!, leaving 7×6.
42. - Six different students line up. How many orders?2 marks
6!=720.
720. - From 6 students choose a captain and deputy. How many choices?2 marks
6×5=30; roles are different.
30. - How many unordered two-person groups can be chosen from 6 students?3 marks
Ordered pairs:6×5=30.
Divide by 2 because each group appears twice.
15. - List the distinct arrangements of AAB and explain why there are not 6.3 marks
Swapping identical A letters does not create a different result.
AAB, ABA, BAA:3 arrangements.
Curriculum reference & lesson scope
Australian Curriculum v 9 / SA Curriculum
Compound events; systematic counting and the multiplication principle; factorials and simple arrangements; conditional language and restricted sample spaces; conditional probabilities from tables/trees; replacement and dependence; conditional versus unconditional probabilities; designing and interpreting simulations.
Official curriculum source ↗