Math 184A First Midterm OPEN BOOK 2/3/99
Each problem is worth 12 points. Please start each problem on a new page.
1. (a) How many four card hands are there where no cards are in the same suit and no
cards are the same kind? (“Kind” means A, 2, 3, ..., 10, J, Q, K.)
(b) How many 6 card hands contain two 3-of-a-kinds; that is, two sets of cards where
each set consists of 3 cards of the same kind?
2. (a) Give an example of a SIMPLE graph with a cut vertex and indicate the cut vertex.
(b) For the graph in (a), indicate a walk which is NOT a path.
(c) Give an example of a SIMPLE graph with 3 vertices and 12 edges.
3. An oriented simple graph is a simple graph which has been converted to a digraph by
assigning an orientation to each edge.
(a) Prove that the number of n-vertex oriented simple graphs is 3(n
2).
(b) State and prove a formula for the number of n-vertex oriented simple graphs that
have exactly qedges.
Hint: You can construct an oriented simple graph by choosing a simple graph
and then orienting each of its edges.
4. The depth of a rooted tree is the number of edges in the longest path from the root
to a leaf. A binary rooted tree is a rooted tree in which each vertex either is a leaf or
has exactly two children. Some examples are on the blackboard.
(a) Let Lnbe the maximum number of leaves in a binary rooted tree of depth n.
Prove that
L0= 1 and Ln=2L
n−1for n>0.
Hint: What happens when the root is removed?
(b) Using (a), prove that Ln=2
nfor n≥0.
(c) Let lnbe the minimum number of leaves in a binary rooted tree of depth n. Prove
that ln=n+1.
