Homework 1 Solutions
Question 1 (Fibonacci Numbers, 30 points). The Fibonacci Numbers is the sequence 1, 1, 2, 3, 5, 8,
13 . . . defined as follows:
F1 = F2 = 1
F n+2 = F n+1 + F n for all n 1
(a) Prove that F n+2 counts the number of strings of length n
Math 184A Homework 2 Solution
Question 1 (Colored Balls in Bins, 30 points). Consider the problem of placing unlabelled red and blue
colored balls into labelled bins. As the balls are unlabelled you cannot distinguish one from the other except
by color. Y
Homework 4 Solutions
Question 1 (Stirling Number Computation). Compute the values of c(n, k) for all 1 k n 6.
Answer: c(n, n) = 1, since the only way to split [n] into n cycles is to have each cycle a
singleton, which gives the identity permutation. From
MATH 184A
HW3 Solutions
1.a. By relabeling we can assume that a1 = n. We then consider a permutation in
which a1 , . . . , ak are in the same cycle. When we put this permutation into canonical
cycle notation, the cycle containing a1 , . . . , ak will star
Math184A HW5
Kevin Chen
Q1:
Prove the LHS & RHS is equal. For LHS, =(, )c(k,m)c(n-k,m)
Lets say we have n element and we marked k<=n elements. The way to pick marked elements
is (n,k). Then we combined k marked circle element to m cycles. Because the numb
Math184 hw3
Kevin Chen
A99016503
1.
a) (k-1)! (n-k)!
Since a1 = n, n = beginning of last cycle. (The biggest number)
b) (n-1-b)!(b-1)!(a-1)!
I = size a, j = size b
I = n, j = n-1 n = a+1th
n-a-b+1th n-1/b (second to the last cycles length is b)
n/a (lengt
Math 184A: Fall 2013
Homework 1 Solutions
Due 5:00pm on Friday 10/4/2013
Problem 1: (Exercise 1.26 in Bna) We are given 17 points inside a regular triangle
o
of side length one. Prove that there are two points among them whose distance is not
more than 1/
Math 184A: Fall 2013
Homework 3
Due 5:00pm on Friday 10/18/2013
Problem 1: (Exercise 4.33 in Bna) Prove, by a combinatorial argument, that for all
o
3n
positive integers n, the number n,n,n is divisible by 6.
Problem 2: (Exercise 4.41 in Bna) Prove that f
Math 184A: Fall 2013
Homework 3 Solutions
Due 5:00pm on Friday 10/18/2013
Problem 1: (Exercise 4.33 in Bna) Prove, by a combinatorial argument, that for all
o
3n
positive integers n, the number n,n,n is divisible by 6.
3n
Solution: The multinomial coecien
Math 184A: Fall 2013
Homework 2
Due 5:00pm on Friday 10/11/2013
Problem 1: (Exercise 3.30 in Bna) How many four-digit positive integers are there
o
in which all digits are dierent?
Problem 2: (Exercise 3.37 in Bna) In a certain year, the month of January
Math 184A: Fall 2013
Homework 1
Due 1:00pm on Friday 10/4/2013
Problem 1: (Exercise 1.26 in Bna) We are given 17 points inside a regular triangle
o
of side length one. Prove that there are two points among them whose distance is not
more than 1/4.
Problem
Math 184A Homework 5 Solution
Question 1 (Combinatorial Identity, 20 points). Come up with a combinatorial proof of the following identity
for n 2m > 0:
nm
X n
2m
c(k, m)c(n k, m) =
c(n, 2m).
k
m
k=m
Solution. We count the same number twice to prove thi
Math 184A, Winter 2017, Prof. Tesler
Homework #5, Due Wednesday February 15, 2017
Note: If you would like to use your homework to study for the exam, please photocopy
it before turning it in.
Chapter 7# 3, 4, 7, 19, 24, 25, 26, 28
Chapter 8# 12, 39
and th
Math 184A Final Exam
Fall 2015
Instructions: Do not open until the exam starts. The exam will run for 180 minutes. The problems are
roughly sorted in increasing order of diculty. Answer all questions completely. You are free to make use
of any result in t
Math 184A Exam 2
Fall 2015
Instructions: Do not open until the exam starts. The exam will run for 45 minutes. The problems are
roughly sorted in increasing order of diculty. Answer all questions completely. In particular, in order to
get full credit, you
Math 184A Homework 4
Fall 2015
This homework is due Monday November 2nd in discussion section. Remember to justify your work even
A
if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Optional Practi
Math 184A Homework 5
Fall 2015
This homework is due Monday November 9th in discussion section. Remember to justify your work even
A
if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Optional Practi
Math 184A Homework 6
Fall 2015
This homework is due Monday November 23rd in discussion section. Remember to justify your work
A
even if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not
required. If you cannot
Math 184A Homework 3
Fall 2015
This homework is due Monday October 26th in discussion section. Remember to justify your work even
A
if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Question 1 (Die
Math 184A Exam 1
Fall 2015
Instructions: Do not open until the exam starts. The exam will run for 45 minutes. The problems are
roughly sorted in increasing order of diculty. Answer all questions completely. In particular, in order to
get full credit, you
Math 184A Homework 1
Fall 2015
This homework is due Monday October 5th in discussion section. Remember to justify your work even if
A
the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Question 1 (Summ
Math 184A Homework 2
Fall 2015
This homework is due Monday October 12th in discussion section. Remember to justify your work even
A
if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Question 1 (Equ
Math 184A HW 2 Solution
Question 1 (Equal Sum Subsets, 20 points). Let S be a set of 10 positive integers each at most 100. Show
that there exist two dierent subsets A S and B S so that the sum of the elements of A equals the sum
of the elements of B.
Ans
Math 184A Homework 1
Fall 2015
This homework is due Monday October 5th in discussion section. Remember to justify your work even if
A
the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Question 1 (Summ
Math 184A, Winter 2017, Prof. Tesler
Homework #4, Due Wednesday February 8, 2017
Chapter 5# 32 (B(n) is the Bell number, defined on page 97; in class, we used notation Bn .)
Chapter 8# 2 , 7, 23, 35
and the problems below, H-9 through H-12 (see both pages
Math 184A, Winter 2017, Prof. Tesler
Homework #3, Due Wednesday February 1, 2017
and the problems below, H-3 through H-8 (see both pages).
Chapter 5# 6, 8, 12 , 26, 27;
Note: On Chapter 5# 12, your answer should be in terms of the function p(n); you dont
Math 184A: Fall 2013
Homework 2 Solutions
Due 5:00pm on Friday 10/11/2013
Problem 1: (Exercise 3.30 in Bna) How many four-digit positive integers are there
o
in which all digits are dierent?
Solution: We can think of a four-digit positive integers as a st