By Herbert S. Wilf

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Ways to form the whole sequence. Of the 2n subsets of [n], how many have exactly k objects in them? The number of elements in a set is called its cardinality. The cardinality of a set S is denoted by |S|, so, for example, |[6]| = 6. ’ The question is, for how many subsets S of [n] is it true that |S| = k? 5. Counting 35 We can construct k-subsets S of [n] (written ‘S ⊆ [n]’) as follows. Choose an element a1 (n possible choices). , until a sequence of k diﬀerent elements has been chosen. Obviously there were n(n−1)(n−2) · · · (n−k+1) ways in which we might have chosen that sequence, so the number of ways to choose an (ordered) sequence of k elements from [n] is: n(n − 1)(n − 2) · · · (n − k + 1) = n!

Most counting problems on graphs are much easier for labeled than for unlabeled graphs. Consider the following question: How many graphs are there that have exactly n vertices? Suppose first¡ that we mean labeled graphs. A graph of n vertices has ¢ a maximum of n2 edges. To construct a graph, we would decide which ¡ ¢ of these possible edges would be used. We can make each of these n2 decisions independently, and for every way of deciding where to put the edges, we would get a diﬀerent graph. Therefore, the number of labeled n graphs of n vertices is 2( 2 ) = 2n(n−1)/2 .

5 2n , (n + 4)12 . 9. Find a function f (x) such that f (x) = O(x1+² ) is true for every ² > 0, but for which it is not true that f (x) = O(x). 10. 2 Positional Number Systems This section will provide a brief review of the representation of numbers in diﬀerent bases. The usual decimal system represents numbers by using the 20 1. Mathematical Preliminaries digits 0, 1, . . , 9. For the purpose of representing whole numbers, we can imagine that the powers of 10 are displayed before us like this: .