MATH 15B REVIEW PROBLEMS
December 2000
1
MATH 15B REVIEW PROBLEMS
Disclaimer: These problems were written by me [Nick Loehr] and represent types of problems that can be solved based on the material Dr. Bender covered in 15B. The subject matter and
Math 15B
Final Exam
8 December 2000
Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but calculators ARE allowed. You must show your work to receive credit. 1. (60 pts) In each case, give an example or explain why
Math 15B
Final Exam Solutions
8 December 2000
1. (60 pts) In each case, give an example or explain why none exists. (a) A tree with exactly nine vertices and exactly nine edges. Ans: Impossible since a tree always has one less edge than it has ver
Math 15B (Bender)
Midterm Exam
27 October 2000
Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but calculators ARE allowed. You must show your work to receive credit.
1. (36 pts) A ve digit number is a sequence o
Math 15B (Bender)
Midterm Exam
27 October 2000
1. (36 pts) A ve digit number is a sequence of ve digits, the rst of which is NOT zero. Thus, 12345 and 10101 are valid but 01234 and 1234 are NOT valid. (a) How many ve digit numbers are there? Ans:
15B Weekly Quizzes Q1. A family has 4 girls and 3 boys. (a) How many ways can they sit in a row? (b) How many ways can they sit in a row if boys and girls must alternate?
Fall 2000
Q2. How many ways can t teams each of size s be made from st people
15B Weekly Quizzes
Fall 2000
Q1. A family has 4 girls and 3 boys. (a) How many ways can they sit in a row? Ans: 7!. (b) How many ways can they sit in a row if boys and girls must alternate? Ans: 4! 3! since the only possible seating pattern in GBG
Math 15B
Final Exam
6 December 2001
Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but TWO PAGES OF NOTES ARE ALLOWED. Calculators are NOT ALLOWED. You need not evaluate binomial coecients. You must show your wo
Math 15B
Final Exam Solutions
6 December 2001
1. In each case, give an example or explain why none exists. (a) A permutation f of {1, 2, 3, 4, 5} such that, for some x {1, 2, 3, 4, 5}, f 20 (x) = x. A. In cycle form, choose f to have a 3-cycle an
Math 15B
Midterm Exam
26 October 2001
Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but a PAGE OF NOTES IS ALLOWED. Calculators are NOT ALLOWED. You need not evaluate binomial coecients. You must show your work
Math 15B
Midterm Exam Solutions
26 October 2001
1. Give an example of each of the following or explain why it cannot be done. (a) A bijection from {1, 2, 3, 4} to {a, b, c}. A. Impossible. A bijection has its range and domain the same size. (b) A
15B Weekly Quizzes
Fall 2001
You may leave sums, products, factorials, binomial coecients, and so on in your answers. Q1. A family has 3 girls and 3 boys. Give reasons for your answers. (a) How many ways can they sit around a circular table? (b) Ho
15B Weekly Quizzes
Solutions
Fall 2001
Q1. A family has 3 girls and 3 boys.
(a) How many ways can they sit around a circular table?
Ans: The answer is 5!. As derived in class and in the text, the number of circular
arrangements of n people is (n 1)!. We h