#2             92.  FREE FOR ALL           

Eight university students decided to celebrate their graduation by having dinner at a restaurant. Seated around the large table, they began to argue about their order of seating. Should it be done alphabetically, according to age, or in order of marks obtained? The discussion became heated and the dinner became cold.

Finally, the waiter said, “Listen my young friends. Let one of you write down the order in which you are now sitting. Return here tomorrow and sit in a different order. The next day do it in yet a different way, and so on until you have tried out all the combinations. When it comes time for you to sit down again in your present order, I promise to serve you a free meal.”

The offer was tempting, and it was decided to meet at the restaurant every day, so as to get the promised free dinner. The waiter got a rise in salary, but did the students get their free dinner?

Hints and strategies

Hint 1                Hint 2              Solution               Extensions
HINT 1

In how many orders can the eight students sit? (Or how many permutations are there?)
HINT 2

Does the fact that the students ar sitting in a circle alter your answer?
SOLUTION

The students may have got their free dinner. Since only the order is important, the number of possibilities is

7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040.

This works out to be nearly 14 years later!  
  
EXTENSIONS

Factorials increase at an alarming rate!

You might like to guess the size of 10! or 20! or 100! etc. Try putting it in concrete terms like number of seconds, or number of centimetres. You wull be surprised.

Try looking up ‘googol’ on Wikipedia to find the approxinmate size of 70!