Counting choices and factorials
A way is one complete possible result. First count how many choices can fill each position; then multiply to count the complete results. The first part of this chapter counts ordered results.
Count choices one position at a time
place 1
4
place 2
3
place 3
2
place 4
1
4 × 3 × 2 × 1 = 4! = 24
Multiply successive choices
Suppose four different books are placed in a row. The first position has 4 choices, the next has 3, then 2, then 1.
Key idea
Use multiplication for a sequence of decisions joined by and then. If one stage has choices and the next has , there are complete outcomes. The number of choices falls only when an object cannot be reused. If reuse is allowed, the number may stay the same.
Worked example
Outfits from separate choices
A student has 3 shirts, 2 pairs of trousers and 4 pairs of shoes. Each outfit uses one of each.
Answer
Factorial shortens a descending product
For a positive whole number :
Symbols
- = positive whole number of distinct objects (none)
- = ways to arrange all n objects (ways)
- .
- . There is one way to select or arrange nothing: leave every position empty.
- Cancel a shared factorial tail before calculating: .
Common mistake
Factorial is not ordinary multiplication by a punctuation mark. is 120, and is 96—not .
Use x! on your calculator
- On most scientific calculators, x! or n! is in the probability menu.
- The calculator is the fastest way to evaluate a large factorial. Write the factorial expression first so your counting method is visible, then evaluate it.
- For a quotient such as , entering is faster and less error-prone.
Your turnIndependent check [2 marks]
Simplify and evaluate .
Show worked answer
Cancel the shared tail.
Answer