BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI
K K BIRLA GOA CAMPUS
Comprehensive Examination (First Semester, 2019-20 )(Closed book)
Course title: Discrete Mathematics (MATH F213)
Date: 06.12.2019 (Friday) Time: 120 minutes Part-A Max. Marks : 46
Instructions:
(1) There are 23 questions. Each question carries 2 marks
(2) Write the correct answer inside the table in the question paper itself.
(4) Answering in pencil and overwriting/scribbling will be treated invalid.
(5) Any rough work should be done at the back of the main answer sheet.
Student’s Name: ID No.:
No. of correct answers No. of wrong answers Total Marks
For Faculty’s use only:
Recheck request:
1. Write the general form of particular solution apn when f (n) = 2n2 +n+1+n2n +3n and the characteristic
polynomial is (r − 2)2 (r − 4)2 (r − 1)3
ANS: c1 n3 + c2 n4 + c3 n5 + (c4 n2 + c5 n3 )2n + c6 3n
2. Find the number of ways a company can assign 7 projects to 4 people so that each person gets at least
one project.
ANS: 47 − 41 37 + 42 27 − 43 17 or 8400
3. Write the general solutions of the recurrence relation an = 6an−1 − 12an−2 + 8an−3
ANS: (c1 + c2 n + c3 n2 )2n
4. Find the generating function for modeling the number of ways to select 9 integers from 1, 2, 3, · · · , n,
no two of which are consecutive. Which coefficient do we want for n = 100
ANS: (1 + x + x2 + · · · )2 (x + x2 + · · · )8 , x91
5. A basket of fruit is being arranged out of manoges, bananas, and oranges. What is the smallest
number of pieces of fruit that should be put in the basket to guarantee that either there are at least
12 mangoes or at least 10 bananas or at least 11 oranges?
ANS: n1 + n2 + n3 − n + 1 = 31
6. Build a generating function for ar , the number of integer solutions to the equation e1 + e2 + e3 + e4 =
r, 2 ≤ ei ≤ 8, e1 even, e2 odd
ANS: (x2 + x4 + x6 + x8 )(x3 + x5 + x7 )(x2 + x3 · · · + x8 )2
K K BIRLA GOA CAMPUS
Comprehensive Examination (First Semester, 2019-20 )(Closed book)
Course title: Discrete Mathematics (MATH F213)
Date: 06.12.2019 (Friday) Time: 120 minutes Part-A Max. Marks : 46
Instructions:
(1) There are 23 questions. Each question carries 2 marks
(2) Write the correct answer inside the table in the question paper itself.
(4) Answering in pencil and overwriting/scribbling will be treated invalid.
(5) Any rough work should be done at the back of the main answer sheet.
Student’s Name: ID No.:
No. of correct answers No. of wrong answers Total Marks
For Faculty’s use only:
Recheck request:
1. Write the general form of particular solution apn when f (n) = 2n2 +n+1+n2n +3n and the characteristic
polynomial is (r − 2)2 (r − 4)2 (r − 1)3
ANS: c1 n3 + c2 n4 + c3 n5 + (c4 n2 + c5 n3 )2n + c6 3n
2. Find the number of ways a company can assign 7 projects to 4 people so that each person gets at least
one project.
ANS: 47 − 41 37 + 42 27 − 43 17 or 8400
3. Write the general solutions of the recurrence relation an = 6an−1 − 12an−2 + 8an−3
ANS: (c1 + c2 n + c3 n2 )2n
4. Find the generating function for modeling the number of ways to select 9 integers from 1, 2, 3, · · · , n,
no two of which are consecutive. Which coefficient do we want for n = 100
ANS: (1 + x + x2 + · · · )2 (x + x2 + · · · )8 , x91
5. A basket of fruit is being arranged out of manoges, bananas, and oranges. What is the smallest
number of pieces of fruit that should be put in the basket to guarantee that either there are at least
12 mangoes or at least 10 bananas or at least 11 oranges?
ANS: n1 + n2 + n3 − n + 1 = 31
6. Build a generating function for ar , the number of integer solutions to the equation e1 + e2 + e3 + e4 =
r, 2 ≤ ei ≤ 8, e1 even, e2 odd
ANS: (x2 + x4 + x6 + x8 )(x3 + x5 + x7 )(x2 + x3 · · · + x8 )2
