EXAMPLE 3.2.5

Gomer has a 20 volume set of World Book Encyclopedia. The 20 volumes are arranged in numerical order. His uncle Aristotle has

challenged him to write down every possible arrangement of the 20 books. Aristotle will pay Gomer $10,000 if he can compete the job within

30 days. The only proviso is that if Gomer doesn't complete the job within 30 days, he will have to pay Aristotle one penny for every

permutation that he has failed to list.

How many different arrangements are there?

The answer is the number of ways to arrange 20 elements: 20 factorial

According to the calculator,

20 factorial = 2.4 times 10 to the eighteenth power.

Gomer is a fast worker. Assuming that he can write down 1 million arrangements per second, how long will it take for him to complete the job?

We divide the previous result by 1 million in order to find the number of seconds:

2.4 times 10 to the eighteenth power divided by one million equals 2.4 times ten to the twelfth power seconds.

Now we convert this to years, one step at a time:

2.4 times ten to the twelfth power seconds divided by 60 seconds per minute equals four times ten to the tenth power minutes.
Four times ten to the tenth power minutes divided by 60 minutes per hour equals 6.7 times ten to the eighth power hours.
6.7 times ten to the eight power hours divided by 24 hours per day equals 2.8 times ten to the seventh power days.
2.8 times ten to the seventh power days divided by 365 days per year equals 77,000 years.

Working at rate of 1,000,000 arrangements per second, it would take Gomer roughly 77,000 years to list every possible arrangement of a set of 20 books.