200852409 - University College Dublin An Cotéista...

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Unformatted text preview: University College, Dublin An Cotéista Qilscuite, Bane Atha Cliath SEMESTER 2 EXAMINATION 2008/2009 MST 10040 Combinatorics and Number Theory Professor P. J. Rippon Professor S. Dineen Dr. T. Unger Dipl. Math. C. Roessing* 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 2008/ 2009M0dular 1 of 2 1. 2. 3. 4. ©UCD 2008 / 2009Modular (a) Expand the summation —- 2). i=1 (b) Use the method of mathematical induction to prove that n _ __ n(3n—— 1) ;(3z — 2) — "T: for all n E N. (c) Calculate the ﬁrst seven terms of the sequence 04, . . . ,am Where n E N, deﬁned by a1 = 2 and the recursion an“ = an + 2 - n. ((1) Use the method of mathematical induction to show that this sequence satisﬁes an = n2 — n + 2 for all n E N. (a) Let n, m E N With m S n. Express in factorial notation. (b)Showthat n n 7" _ n . 11—19 7" k — k r—k Wherelgkgrgn. (c) In Poker each player gets 5 cards from a standard pack of 52 cards. HOW many different hands are possible? How many hands are possible Without all court cards (Ace, King, Queen and Jack)? (a) State the binomial theorem for (x + y)", n E N. (b) Use the binomial theorem to prove that <:)+(:>+~-+(:>=2“- (c) Find the octal (base 8) representation of 2009. Which decimal number has the binary representation 11110101101? (a) State Fermat’s Little Theorem and use it to ﬁnd the smallest positive integer a: satisfying 290 E x mod 47. (b) Find all mutually incongruent solutions of the linear congruence 120 - a: E 63 mod 321. (c) Use the Chinese remainder theorem to ﬁnd the smallest positive integer that satisﬁes the following system of linear congruences: 3 mod 9 2 mod 11 1 mod 13 Ill HI Ill 20f2 ...
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