Combinatorics Through Guided Discovery, 2004a

4 1. What is Combinatorics?
For example, you might start with f (1) = a, f (2) = b. How many
functions are there from the set {1, 2} to the set {a, b}? (h)
(b) How many functions are there from the three element set {1, 2, 3} to
the two element set {a, b}? (h)
(c) How many functions are there from the two element set {a, b} to the
three element set {1, 2, 3}? (h)
(d) How many functions are there from a three element set to a 12 element
set?
(e) The function f is called one-to-one or an injection if whenever x is
different from y, f (x) is different from f (y). How many one-to-one
functions are there from a three element set to a 12 element set?
(f) Explain the relationship between this problem and Problem 6.
• Problem 8. A group of hungry team members in Problem 6 notices it would
be cheaper to buy three pints of ice cream for them to split than to buy a
triple decker cone for each of them, and that way they would get more ice
cream. They ask their coach if they can buy three pints of ice cream.
(a) In how many ways can they choose three pints of different flavors out
of the 12 flavors? (h)
(b) In how many ways may they choose three pints if the flavors don’t
have to be different? (h)
• Problem 9. Two sets are said to be disjoint if they have no elements in
common. For example, {1, 3, 12} and {6, 4, 8, 2} are disjoint, but {1, 3, 12}
and {3, 5, 7} are not. Three or more sets are said to be mutually disjoint if
no two of them have any elements in common. What can you say about the
size of the union of a finite number of finite (mutually) disjoint sets? Does
this have anything to do with any of the previous problems?
Problem 10.
• Disjoint subsets are defined in Problem 9. What can you say
about the size of the union of m (mutually) disjoint sets, each of size n?
Does this have anything to do with any of the previous problems?