2005/06 Undergraduate Module Catalogue
10 creditsClass Size: 200
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2005/06
Pre-requisitesMATH2210 or equivalent.
This module is approved as an Elective
ObjectivesMany counting problems arise in the real world in the form 'how many ?' Not all such problems have easy solutions. This module introduces techniques for dealing with some of the more difficult problems of this kind. On completion of this module, students should be able to decide which of the several ?counting? techniques (e.g. inclusion-exclusion principle, dot diagram method, generating function method etc.), introduced in the course, is best suited to solving any one of a large class of counting problems and should be able to apply that technique to complete the problem?s solution.
Combinatorics is concerned with the study of arrangements, patterns, designs, assignments, schedules, configurations, in particular with the question 'How many ?' Its applications are widespread: computer scientists, bankers, statisticians, works supervisors, industrial engineers being amongst its users. Serious problems even have their more frivolous equivalents which delight the mathematician. For example: (i) How many different solutions are there in positive integers x, y, z, t to the inequality x + y + z + t < 41 ? (ii) If you and I each have a pack of 52 cards what are the chances, if we simultaneously turn over one card at a time, that at least one pair matches exactly ? In how many different ways can you paint the squares of an 8 x 8 grid with black/white paint if two paintings are regarded as the same if one can be obtained from the other by a rotation? To solve these and similar problems is not hard - if you go about it the right way. Topics covered include: 1. Sum Rule, product rule. 2. Problems involving binomial coefficients. Occupancy problems. Inclusion-exclusion principle. Derangements. 3. Partitions. Dot diagrams. 4. Stirling?s formula. 5. Generating functions. Theory and lots of examples. Homogeneous and inhomogeneous linear recurrences (by generating function and characteristic equation method). 6. Polya's theory of counting. Including Burnside's lemma, the orbit stabliliser theorem and Polya's theorem. 7. Dirichlet's box principle. Generalisation to Ramsey numbers.
Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for informationLectures: 20 hours. 6 examples classes.
Methods of assessment
Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for informationOne 2 hour examination at end of semester (100%).
Reading listThe reading list is available from the Library website
Last updated: 13/05/2005
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