Factorials & the 'ways' mindset
- This chapter counts ways. The main move is simple: multiply the number of choices at each stage.
- A factorial is the count for lining up all the objects:.
- . Think of it as the empty job: there is exactly one way to do nothing.
Cancel the tail
Factorials are descending products. When one factorial sits over another, cancel the shared tail before you touch the calculator.
- means .
- leaves only the new part:.
- This is why factorial work is often short. You do not want giant numbers; you want the few factors that survive.
Worked example
Small factorial expressions
- .
- .
Answer
Common mistake
is not . Expand or factor the shared: .
Why 0! is 1
- The factorial list steps down by dividing:, , .
- One more step gives .
- It also makes formulas work: .