Counting Principles NMH Upward Bound Summer Academy Statistics H. Reardon

Fundamental Counting Principle  For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m n ways.  The fundamental counting principle extends to situations involving more than two events.  When you have the same event x happening y times (such as an alarm system code or a padlock), do x y for a shortcut to the total number of ways.

Example  You finished college and are looking to buy your first car. There are two styles: sedan or SUV; five colors: black, red, green, blue, or yellow; and three models: standard, sports, and luxury. How many possible combinations are there?  2 styles 5 cars 3 models = 30 total car choices

Factorial Rule  A collection of n different items can be arranged in order n! different ways.  Example: You have 6 textbooks on a bookshelf. How many different can you arrange these 6 textbooks?  6! = 6 · 5 · 4 · 3 · 2 · 1 = 720 different ways

Permutations

Example

Combinations

Example

License Plates  In California, drivers of normal cars who do not elect to have a custom license plate (which is expensive) will have a license plate assigned to them. The plate consists of 7 places: one digit 0-9, followed by three letters A-Z, followed by three digits 0-9. How many possible license plates are there?  How many ways can you arrange the three letters in the middle?  This only applies to passenger cars. Flatbed trucks have a different system. Motorcycles another. Commercial vans yet another. Why do you think there are so many possible permutations?