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Posted By: sac Member Level: Diamond Posted Date: 28 Apr 2008
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2007 Alagappa University M.Sc Computer Science DISCRETE AND COMBINATORIAL MATHEMATICS Question paper
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DISTANCE EDUCATION
M.Sc. DEGREE EXAMINATION, DECEMBER 2007.
Mathematics
DISCRETE AND COMBINATORIAL MATHEMATICS
Time : Three hours Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a) Prove the identity
in two ways.
(b) Find the numbers of r-digit quaternary sequences that contain an even number of 0’s.
2. (a) Solve the difference equation : .
(b) Find the number of n-digit binary sequences that have exactly r pairs of adjacent 1’s and no adjacent 0’s.
3. (a) Find the number of integers between 1 and 250 that are not divisible by any of the integers 2, 3, 5 and 7.
(b) Define derangement. Find the number of derangements of n objects.
4. (a) In how many ways can the letters be arranged so that all the letters of the same kind are not in the single block?
(b) Explain the permutations with forbidden positions. Define hit polynomial and obtain a formula for it.
5. (a) Find the number of distinct strings of length 2 that are made up of blue beads and yellow beads.
(b) Find the number of all possible ways of painting three distinct balls in solid colors when there are three kinds of paint available, an expensive kind of red paint, a cheap kind of red paint, and a blue paid.
6. (a) Show that the number of equivalence classes of functions from to is given by where is the number of functions of which are such that for all in .
(b) Show further that the above number is the value of the expression evaluated at .
7. (a) Show that the lattice is modular but not distributive and that is not modular.
(b) Show that a lattice is distributive if and only if it has no sublattices isomorphic to or
8. (a) Define Boolean expressions, Minterm, maxterm, Hasse diagram and Canonical forms.
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