LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
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B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – November 2008
MT 5402 – COMBINATORICS
Date : 14-11-08 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
SECTION-A
ANSWER ALL THE QUESTIONS: (10×2 = 20)
- Define Stirling number of second kind. Find
- How many ways can one move from (0,0,0 ) to (a,b,c).
- Find per .
- Find the cycle of length 1, length 3 and length 4 for the permutation 1 2 3 4 5 6 7 8 3 2 5 1 4 8 6 7 .
- Define Ordinary generating function.
- Define Ferrers graph.
- Find the Rook polynomial for the chess board C in the diagram below,
- Define Euler’s function.
- Define derangement using Sieve’s formula.
- Define cycle index of a permutation group.
SECTION-B
ANSWER ANY FIVE QUESTIONS: (5×8 = 40)
- An executive attending a week long business seminar has five suits of different colours. On Mondays, she does not wear blue or green, on Tuesdays, red or green, on Wednesdays, blue, white or yellow, on Fridays, white. How many ways can she dress without repeating a colour during the seminar?
- Prove that the element f of R [t ] given by f(t) = has an inverse in R[t] if and only if has an inverse in R.
- Prove that the cardinality of the set of permutations of m symbols taken n at a time is m(m-1) (m-2)……..(m-n+1).
- State and prove Multinomial theorem .
- (i) An examination paper with 10 questions consists of 6 questions in algebra and 4 questions in geometry. At least 1 question from each section is to be attempted. In how many ways can this be done?
(ii) Find the number of ways of selecting 4 letters from the word EXAMINATION. (5+3)
- (i) Define exponential generating function.
(ii) How many permutations can be formed from the two symbols and with the added condition that may occur at most twice and may occur at most once. Illustrate the problem using exponential generating function.
(3+5)
- If the number of permutation in of type (λ) = , then show that .
- Describe the following Symmetric functions with an example
(a) Monomial (b) Complete homogenous (c) Elementary.
SECTION-C
ANSWER ANY TWO QUESTIONS (2×20 = 40)
- (i) If there exists a bijection between the set of n-letter words with distinct letters out of an alphabet of m letters and the set of n-tuples on m letters, without repetitions. Show that the cardinality of each of these sets is
m(m-1)(m-2)…(m-n+1).
(ii) Define increasing and strictly increasing of a word. Also prove that there exists a bijection between the set of increasing words of length n on m ordered letters, and the set of distributions on n non-distinct objects into m distinct boxes.
(10+10)
- (i) In how many ways can a total of 16 be obtained by rolling 4 dice once?
(ii) Define a partition for the integer n. Give the recurrence formula for . Tabulate the values of for n, m = 1,2,3…7 (8+12)
- (i) State and prove Generalized Inclusion and Exclusion principle.
(ii) With proper illustrations describe the problem of Fibonacci.
(10+10)
- State and prove Burnside’s lemma.
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