In 1858 Alexander Rhind a Scottish antiquarian visiting Luxor came across a 5
metre long papyrus filled with Egyptian mathematics from the period 1800 BC. He
bought it. A few years later it was acquired by the British Museum and its
hieroglyphics translated. Problem 79 of the Rhind Papyrus is a problem of
houses, cats, mice and wheat very similar to the kits, cats, sacks and wives of St
Ives. Both involve powers of 7 and the same kind of analysis. Combinatorics, it
seems, has a long history.
Factorial numbers
The problem of queues introduces us to the first weapon in the combinatorial
armoury – the factorial number. Suppose Alan, Brian, Charlotte, David, and Ellie
form themselves into a queue
E C A B D
with Ellie at the head of the queue followed by Charlotte, Alan and Brian with
David at the end. By swapping the people around other queues are formed; how
many different queues are possible?
The art of counting in this problem depends on choice. There are 5 choices for
who we place as the first person in the queue, and once this person has been
chosen, there are 4 choices for the second person, and so on. When we come to
the last position there is no choice at all as it can only be filled by the person left
over. There are therefore 5 × 4 × 3 × 2 × 1 = 120 possible queues. If we
started with 6 people, the number of different queues would be 6 × 5 × 4 × 3 ×
2 × 1 = 720 and for 7 people there would be 7 × 6 × 5 × 4 × 3 × 2 × 1 =
5040 possible queues.
A number obtained by multiplying successive whole numbers is called a
factorial number. These occur so often in mathematics that they are written using
the notation 5! (read ‘5 factorial’) instead of 5 × 4 × 3 × 2 × 1. Let’s take a look
at the first few factorials (we’ll define 0! to equal 1). Straightaway, we see that