MA241 Combinatorics Marking Sheet 2
Deadline: Monday, 30 October 2006, 3:00.
For this sheet, (B3) and (B6) will be assessed.
(B3). Find a necessary and sucient condition on the real number b > 1
for the equality
logb x = logb x
to hold for all real x 1.
S
The primary structure of amino acids
Amino term always be on amino acid one
Carboxyl term always on terminal amino acid
Linear no branches
Only called proteins when in a functional state gone through folding with ions on
added post-transcriptional modific
A-HELIXES PART 2- THIS CAN BE USED ALONGSIDE A TEXTBOOK FOR INFORMATION
Amino term always be on amino acid one
Carboxyl term always on terminal amino acid
Linear no branches
Only called proteins when in a functional state gone through folding with ions on
The primary structure of amino acids
Amino term always be on amino acid one
Carboxyl term always on terminal amino acid
Linear no branches
Only called proteins when in a functional state gone through folding with ions on
added post-transcriptional modific
Career Committees
Engineering and
STEM event 2015
Tuesday 10th February
4:30-6:00pm, Lecture Theatre
4:30-4:35 Introduction of
speakers
4:35-4:55 Tao Cheng, Civil
Engineering Professor at
University College London
4:55-5:10 Peter Childs, Head of
Design En
MA241 Combinatorics Sheet 3
Deadline: Monday, 13 November 2006, 3:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA241 Combinatorics box in front of the Undergraduate Oce.
(A1) In the lectures we unfolded the basic recur
MA241 Combinatorics Sheet 2
Deadline: Monday, 30 October 2006, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA241 Combinatorics box in front of the Undergraduate Oce.
(B1) Let n 0. Express Tn =
5
k
in terms of n and
MA241 Combinatorics Sheet 5
Deadline: Monday, 15 January 2007, 3:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA241 Combinatorics box in front of the Undergraduate Oce.
m
m+1
Bj = [m = 0].
j
(A1) Use binomial convoluti
MA241 Combinatorics Marking Sheet 5
Deadline: Monday, 15 January 2007, 3:00.
For this sheet, B3(a) and B6 will be assessed.
(B3).
(a) Let t R. Find a closed formula for the exponential generating function
for the numbers an dened by
n
a0 = 1,
an+1 = t
k=0
MA241 Combinatorics Marking Sheet 1
Deadline: Monday, 23 October 2006, 2:00.
For this sheet, (B4)(b) and (B10) will be assessed.
(B4)(b). Use the summation factor method to solve the following, that is,
to nd a closed formula for bn (n 0).
(b) b0 = 0, bn
1
Introduction: 3 Sample Problems
Example 1: Tower of Hanoi.
n discs
A
B
C
Object: Move pile A to B by moving one disc at a time. A disc may never
rest on a smaller one. What is the minimal number of moves, Tn ?
Strategy: for n > 1
1. Move top (n 1) discs
1. cfw_n is one of the subsets. Then there are
Xn \ cfw_n into k 1 non empty subsets.
n1
k1
ways of decomposing
2. n is in a larger subset. Then we decompose Xn \ cfw_n into k non-empty
subsets in n1 ways. We can place n in any of these k subsets. So
k
th
MA241 Combinatorics Sheet 1
Deadline: Monday, 23 October 2006, 2:00.
Solutions to Section B are for handing in. Section A is easier and is meant
as a warming up. Questions in section C are similar to those in section B
but not for handing in. Section D is
MA241 Combinatorics Marking Sheet 3
Deadline: Monday, 13 November 2006, 3:00.
For this sheet, (B4) and (B8) are marked.
(B4). Let 0 n. Prove:
n+1
=
+1
k
n
k
k
.
Hint: Apply part 4 of Theorem 12 to x and x + 1 and use the binomial
theorem.
Solution. [15 ma
MA241 Combinatorics Marking Sheet 4
Deadline: Monday, 27 November 2006, 3:00.
For this sheet, B1 (10 marks in total) and B4b (15 marks in total) will be
assessed.
(B1). Compute the generating function of (a0 , a1 , . . .) dened by a0 = 0,
a1 = 1 and
an =
THE UNIVERSITY OF WARWICK SECOND YEAR EXAMINATION: 2004-2005 Mock exam MA241 COMBINATORICS
Time Allowed: 2 hours Read carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book. Calculators ar
MA241 Combinatorics Sheet 4
Deadline: Monday, 27 November 2006, 3:00. Solutions to Section B are for handing in. Please put your solutions into the MA241 Combinatorics box in front of the Undergraduate Office. (A1) Write down closed formulas for the gener
A-HELIXES PART 2- THIS CAN BE USED ALONGSIDE A TEXTBOOK FOR INFORMATION
Amino term always be on amino acid one
Carboxyl term always on terminal amino acid
Linear no branches
Only called proteins when in a functional state gone through folding with ions on