# 200953621 - University College Dublin An Coiéiste(Jimmie...

• Notes
• 2

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: University College Dublin An Coiéiste (Jimmie. Baits Atha Cliath SEMESTER 1 EXAMINATION 2009/2010 MST 10040 Combinatorics and Number Theory Professor P. J. Rippon Dr. Micheal o Searcoid Dr. T. Unger Dipl. Math. C. Roﬁing* Time Allowed: 2 hours Instructions for Candidates Full marks will be awarded for complete answers to all four questions. Instructions for Invigilators Non—programmable calculators may be used during this examination. ©UCD 2009/ 10/ Modular 1 of 2 \$362! n 1. (a) Expand the summation Z 2i. i=1 (b) Use the method of mathematical induction to prove that V: zi = 2”+1 ~ 2 i=1 ' for all n E N. (c) Find the ﬁrst seven terms of the sequence a1, a2, . . . ,an where 0.1 = 3 and an = 7 ' an_1. (d) Proof by mathematical induction that an = 3 - 7”“1 for all n E N. 2. (a) Let 71,7" 6 N with r S n. Express in factorial notation. (b) Showthat n _ n—r n r+1 _ r+1 7“ I (c) How many license plates consist of from zero to three letters followed by from zero to four digits where the all blank plate is not allowed? Suppose 85 letter combinations are not allowed because of their potential for giving offense, how many different plates are possible? (d) In a simple language there are the usual 26 letters, but all words have exactly 4 letters. Any arrangement of the letters, including repetition, is allowed. How many words are there? How many words would that language have if there were also 1, 2 and 3 letter words allowed? 3. (a) State the binomial theorem for (a: + y)”, n E N. (b) Use the binomial theorem to ﬁnd the coefﬁcient of a6b6 in the expansion of (a2 + b3)5. (0) Which decimal number has the octal representation (123)3? What is the hexadecimal (base 16) representation of (2009)10? 4. (a) Use Fermat’s Little Theorem to ﬁnd the smallest positive integer n sat— isfying 290 E n mod 47. (b) Find all mutually incongruent solutions of the linear congruence 142 t as E 16 mod 410. (c) Use the Chinese remainder theorem to ﬁnd the smallest positive integer satisfying the following system of linear congruences: 4mod5 6mod7 2mod6 ll! ©UCD 2009/lO/Modular 2 of 2 ...
View Full Document

• Fall '13

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern