Math 475, Problem Set #13: Answers A. A baton is divided into ve cylindrical bands of equal length, as shown (crudely) below. _ _ _ _ _ ()_)_)_)_)_) In how many dierent ways can the ve bands be colored if n colors are available, and unlimited repetition o
Math 475, Problem Set #1: Solutions
A. Section 1.8, problem 3. (Hint #1: Use a coloring-argument, as in section 1.1. Hint #2: Try playing with smaller even-by-even rectangles if 8-by-8 seems too big.) The prisoner cannot escape. Imagine coloring the cells
Anthony Saliva
MA1310: Module 1 Sequences
1. a1 =13
a2 = 2nd term
a2 = a21 +8 = 13+8 = 21
a3 = 3rd term
a3 = a31 +8 = 21+8 = 29
a 4 = 4th term
a 4 = a 41 +8 = 29+8 = 37
The first four terms are 13, 21, 29, and 37
2.
16 !
2 ! 14 !
16 x 15 x 14
2 x 1 x 14
1
Math 475, Problem Set #11: Answers
A. Chapter 8, problem 2. We can put these arrays into one-to-one correspondence with acceptable sequences of +1's and -1's. Given such an array, define ak (for all k between 1 and 2n) to be +1 if k appears in the first r
Math 475, Problem Set #6: Solutions
A. (a) For each point (a, b) with a, b non-negative integers satisfying a+b 8, count the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i + 2, j), (i, j + 2), and (i + 1, j + 1). Ill do this in the
Math 475, Problem Set #10: Answers
A. Consider the sequence 1, 2, 8, 40, 224, 1344, 8448, 54912, . . . dened by the initial condition a1 = 1 and the recurrence relation an = 2(a1 an1 + a2 an2 + . . . + an1 a1 ) (valid for all n 2). Find (and prove) a gene
Math 475, Problem Set #5: Solutions A. Chapter 3, problem 28. Do part (a) in two dierent ways: once by brute force (i.e., dynamic programming), and once by interpreting the counting of routes in terms of multiset permutations. Likewise, do part (b) in two
Math 475, Problem Set #3: Solutions
A. Section 3.6, problem 1. Also: How many of the four-digit numbers being considered satisfy (a) but not (b)? How many satisfy (b) but not (a)? How many satisfy neither (a) nor (b)? No constraints: 5 5 5 5 = 625 (by the
Math 475, Problem Set #2 (due 2/5/04) A. Section 2.4, problem 5. Divide the integers from 1 to 3n into n triples: 1 through 3, 4 through 6, 7 through 9, etc. If n + 1 integers between 1 and 3n are chosen, two of them must lie in the same triple; but these
Math 475, Problem Set #4: Solutions
A. Chapter 3, problem 30. If the person sitting to the right of the dog is a man, the people must alternate in gender thereafter, and there are 5! ways to assign the men to the five seats available for men and 5! ways t
American National Standards Institute, this institute creates the engineering standards
for North America. The ANSI or more correctly ASME Y14.5-2009 for dimensioning and
tolerancing is in fact a metric standard. The ASME Standard for format size etc. is
Solutions to Midterm Exam 1. Use a parity argument to show that there does not exist a path from (3, 3) to (3, 3) (travelling by unit-steps in the +x, x, +y, and y directions) that avoids the point (0, 0) and visits each of the other lattice points cfw_(i