Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2008: Test 2
Name:
Problem
1
2
3
Total
Points
33
33
34
100
1
Score
Q1: (33pts)
The sequence a0 , a1 , . . . , an , . . . satises the following:
a0 = 1 and
an 4an1 = 3n
for n 1
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2008: Test 1
Name:
Problem
1
2
3
Total
Points
33
33
34
100
1
Score
Q1: (33pts)
Show that the number of ways of placing k as and n k bs or cs on the
vertices of an n vertex pol
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2007: Test 4
Name:
Problem
1
2
3
Total
Points
33
33
34
100
1
Score
Q1: (33pts)
Consider the following two one pile take-away games.
(a): In game one you can take away 1,2 or 3
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2007: Test 3
Name:
Problem
1
2
3
Total
Points Score
33
33
34
100
1
Q1: (33pts)
Let N = n4 + 1 and let a1 , a2 , . . . , aN and b1 , b2 , . . . , bN be two sequences of
real nu
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2007: Test 2
Name:
Problem
1
2
3
Total
Points Score
33
33
34
100
1
Q1: (30pts)
A Hamilton path in a tournament on vertex set [n] is a permutation of [n]
such that (i + 1) beat
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2007: Test 1
Name:
Problem
1
2
3
Total
Points Score
33
33
34
100
1
Q1: (33pts)
Show that the number of ways of placing k 1s and n k 0s on the vertices of
an n vertex polygon s
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2006: Test 4
Name:
Problem
1
2
3
Total
Points Score
33
33
34
100
1
Q1: (33pts) A is an n n matrix with entries 0 or 1. Let k be a positive
integer. Show that if n R(2k, 2k ) t
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2006: Test 3
Name:
Problem
1
2
3
1 Total
Points Score
33
33
34
100
1
Q1: (33pts)
(a) A connected graph G has two paths P1 , P2 of lengths k1 , k2 respectively.
Show that if P1
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2006: Test 2
Name:
Problem
1
2
3
1 Total
Points Score
33
33
34
100
1
Q1: (33pts) A box has m drawers. Drawer i contains gi gold coins and
si silver coins where gi + si 2, for
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2006: Test 1
Name:
Problem
1
2
3
Total
Points Score
33
33
34
100
1
Q1: (33pts)
(a): Given integers m, n > 0 and an integer a 0, show that the number of
functions f from [n] to