Understand the idea.
Factorials count arrangements of distinct objects
For a positive integer , factorial notation abbreviates the descending product from to . Arranging distinct objects in a row produces this product because every filled position leaves one fewer object available.
This course uses factorials of non-negative integers. The convention counts one empty arrangement: there is exactly one way to put no objects into no positions. It also keeps counting formulas valid when every object is selected.
Cancel shared factors before evaluating
To simplify a factorial fraction, expand the larger factorial only as far as the smaller one. For instance, exposes the common factor without evaluating either large factorial.
Cancel common factors by division, then calculate the remaining product. A factorial sign applies to the whole number immediately before it; subtracting factorial arguments or cancelling individual digits is not a valid shortcut.
See the method in action.
Arrange a display
Six different postcards are placed in a row. Count the arrangements.
Worked solution
- 1
Count the choices by position.
The postcards are distinct and no postcard can occupy two positions.
- 2
Express the descending product.
Every row is determined by filling all six positions in order.
- 3
Evaluate the product.
The multiplication principle counts each distinct postcard row once.
720.
Simplify first
Evaluate .
Worked solution
- 1
Expose the common factorial.
The descending product from to is exactly the denominator factor .
- 2
Cancel the common factor.
Dividing numerator and denominator by the nonzero factor leaves the ratio unchanged.
- 3
Evaluate the remaining factorial and fraction.
The remaining denominator is , which equals .
36.
Try it yourself.
Check your understanding
What is ?
Show answer & reasoning
20.
Write . Dividing by leaves .
Keep practising
6 worksheet questions · 13 marks
Read the questions online
- Evaluate .2 marks
- Simplify .2 marks
- Evaluate .2 marks
- How many orders can 5 different runners finish in, assuming no ties?2 marks
- Solve for a non-negative integer .2 marks
- Explain why is 1 rather than 0 in the expression .3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Evaluate .2 marks
.
24. - Simplify .2 marks
.
56. - Evaluate .2 marks
.
21. - How many orders can 5 different runners finish in, assuming no ties?2 marks
.
120. - Solve for a non-negative integer .2 marks
, , and factorials increase for integers at least 2.
. - Explain why is 1 rather than 0 in the expression .3 marks
With , .
Using zero would cause division by zero and break the counting interpretation.
Choosing all 5 objects should have one possible selection, so the expression must be 1.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.
Official curriculum source ↗