The fundamental counting principle, factorials, permutations of n objects taken r at a time, and permutations with repeated objects.
Counting without listing
How many license plates are possible? How many ways can a race finish? Listing every possibility would take forever. Counting techniques get the number directly.
The fundamental counting principle
Factorials and permutations
A permutation is an arrangement in which order matters.
Six different books can be arranged on a shelf in \(6! = 720\) ways.
Permutations with repeated objects
If some objects are identical, swapping them doesn't create a new arrangement. Divide by the factorial of each repeat count.
Practice
Quick check
What is \(\frac{8!}{6!}\)?
\(\frac{8!}{6!} = 8 \cdot 7 = 56\).
Quick check
A combination lock has \(3\) dials, each with digits \(0\) to \(9\). How many settings are there?
\(10 \cdot 10 \cdot 10 = 1000\).
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