14 Combinatorics Arithmetic begins with counting, and throughout all branches of mathematics, counting continues to be important. How many possible Tasmanian car number plates with the patternletter + letter + number + number + number + numb are there? How many different hydrocarbons have a chain of20 carbon atoms? Counting is particularly important in probability. What is the probability that a five-digit number without any is divisible by 3? This chapter introduces some of the standard methods of counting and applies them to calculating probabilities. Counting ordered selections with repetition is easily done using powers. Counting permutations - ordered selections without repetition - requires the use of factorials such as 4! = 4 x 3 x 2 x 1 and the symbol"P ... Counting unordered selections requires the use of the symbol"C,, also written as (2). The next chapter examines binomial expansions(x + y)". It turns out that basic counting methods, and particularly the numbers'C',, are essential for understanding these expansions. Digital Resources are available for this chapter in the Interactive Textbook and Online Teaching Suite. See the overview at the front of the textbook for details.
14A Factorial notation 665 14A Factorial notation How many ways can five people form a queue at a bus stop? Solutions of such problems will be formalised in the next section as the multiplication principle, but this particular problem can be solved straightforwardly, and the boxes below help to visualise the solution. · Choose the person at the head of the queue in five ways. · Four people remain, so the person in second place can be chosen in four ways. · Three people remain, so the person in third place can be chosen in three ways. · Two people remain, so the person in fourth place can be chosen in two ways. · And there is now only one way to choose the person in the last place. 1 st place 5 2nd place 4 3rd place 3 4th place 2 5th place 1 Number of possible queues = 5 × 4 × 3 × 2 × 1. What is the fifth derivative of y = x5? Differentiating five times, y' = 5x4 y" = 5 x 4 x x3 y" = 5 x 4 x 3 x x2 y (4) . =5×4×3×2Xx y(5) =5×4×3×2×1. Such products of descending sequences of whole numbers occur in many areas of mathematics, particularly combinatorics, and it is useful to have a name and a notation for them. The product above is written as5! and is called 5 factorial, 5! = 5 × 4 × 3 x 2 x 1 (which evaluates to 120). The definition ofn factorial Informally, the number n! is thought of as the product of all the positive whole numbers from n down to 1: n! = n × (n - 1) × (n - 2) × ··· × 2×1. But the better definition of n! is recursive. First define 0! and then for each successive value of n, say exactly how