Combinatorics Through Guided Discovery, 2004a

8 1. What is Combinatorics?
+ Problem 16. How does the general product principle apply to Problem 6?
• Problem 17. In how many ways can we pass out k distinct pieces of fruit to
n children (with no restriction on how many pieces of fruit a child may get)?
• Problem 18. How many subsets does a set S with n elements have? (h)
◦ Problem 19. Assuming k ≤ n, in how many ways can we pass out k distinct
pieces of fruit to n children if each child may get at most one? What is the
number if k > n? Assume for both questions that we pass out all the fruit. (h)
• Problem 20. Another name for a list, in a specific order, of k distinct things
chosen from a set S is a k-element permutation of S. We can also think of
a k-element permutation of S as a one-to-one function (or, in other words,
injection) from [k] ={1, 2,...,k} to S. How many k-element permutations
does an n-element set have? (For this problem it is natural to assume k ≤ n.
However the question makes sense even if k > n. What is the number of
k-element permutations of an n-element set if k > n? (h)
There are a number of different notations for the number of k-element permutations
of an n-element set. The one we shall use was introduced by Don Knuth;
namely n k , read “n to the k falling” or “n to the k down”. In Problem 20 you may
have shown that
k∏
n k = n(n − 1) ···(n − k +1)= (n − i +1). (1.1)
It is standard to call n k the k-th falling factorial power of n, which explains
why we use exponential notation. Of course we call it a factorial power since n n =
n(n − 1) ···1 which we call n-factorial and denote by n!. If you are unfamiliar with
the Π notation, or product notation we introduced for products in Equation (1.1),
it works just like the Σ notation works for summations.
i=1
• Problem 21. Express n k as a quotient of factorials.
⇒ Problem 22. How should we define n 0 ?