Training Camp

In Warm Up , the total number of starting formations for the volleyball players was written as 4 x 3 x 2 x 1. This string of numbers can be written in shorthand as 4! and is read four factorial . When working with factorial notation, you refer to the string of numbers 4 x 3 x 2 x 1 as expanded form .

Later in this unit, you will use factorials to solve various counting problems. In this lesson you will explore factorial notation in detail.

When working with factorials, you will find that unbelievably large numbers are sometimes the answers to innocent looking questions. For instance, imagine that you are playing with an ordinary deck of 52 cards. As you shuffle and re-shuffle the deck you wonder, how many ways can the deck be shuffled ? That is, how many different ways can the deck be put in order ? You reason that there are 52 choices for the first card then, 51 choices for the second card then, 50 choices for the third card, and so on. This gives a total of 52! ways to order a deck of cards. This number turns out to be a 68-digit monster!

52! = 80658175170943878571660636856403766975289505440883277824000000000000

 

This means that if everyone on earth shuffled cards at a rate of 1000 shuffles per second from now until the end of the millennium, we would not even scratch the surface of getting all possible orders for the deck of cards. No wonder mathematicians developed the shorthand 52!.

Look more closely at the definition of factorial notation.

The definition says factorials are defined only for natural numbers . This means that a string of numbers that are not natural numbers cannot be written using factorial notation. In addition, the definition specifies that factorial notation must have numbers that are consecutive and descending . This means that in factorial notation you can use only natural numbers written one after another in order from greatest to smallest.