Jain College of Engineering, Belagavi

Department of Mathematics
Module 2: Properties of the Integers
Assignment – 2
Course Name: Discrete Mathematical Structures Course Code: BCS405A
Semester : IV Maximum Marks : 20
Date: 12/03/2025 Date of Submission:
Q.NO Question CO PO RBT
State well ordering principle and Prove that
1 � 2�+1 (2�−1) 2 1 L1
12 + 32 + 52 +∙∙∙∙∙∙∙∙∙+ (2� − 1)2 = by Mathematical Induction.
3
By Mathematical Induction Prove that
� �+1 (2�+7)
i) 2 i) 1.3 + 2.4 +∙∙∙∙∙∙∙∙∙∙+ � � + 2 = 6
ii) ��=1 �2� = 2 + (� − 1)2�+1 2 1 L1
� 1 �
iii) �=1 �(�+1)
= �+1
Prove by mathematical induction that, for every positive integer n,
3 2 1 L1
5 divides �2 − �.
4 Prove that every positive integer n≥24 can be written as a sum of 5’s and or 7’s. 2 1 L1
Let �0 = 1, �1 = 2, �2 = 3 and �� = ��−1 + ��−2 + ��−3 for � ≥ 3 . Prove that
5 2 1 L1
�� ≤ 3� ∀� ∈ �+
a) Find recursive definition for the sequence �� ,
i) �� = 5� ii) �� = 2 − −1 �
6 2 1 L1
b) Find explicit definition of the sequence defined recursively by
i) �1 = 4, �� = ��−1 + � ��� � ≥ 2. ii) �1 = 7, �� = 2��−1 + 1 ��� � ≥ 2.
A sequence �� is defined recursively
7 �0 = 1, �1 = 1, �2 = 1, �� = ��−1 + ��−3 ��� ��� � ≥ 3. 2 1 L1
�
Prove that ��+2 ≥ 2 for all integer, � ≥ 0.
Define sum rule and product rule. A bit is either 0 �� 1. A byte is a sequence of 8
bits. Find i) the number of bytes, ii) the number of bytes that begins with 11 and
8 2 1 L1
ends with 11, iii) the number of bytes that begins with 11 and do not ends with 11,
iv) the number of bytes that begins with 11 or ends with 11.
How many positive integers n can be formed using the digits 3,4,4,5,5,6,7 if n is to
9 2 1 L1
exceed 50,00,000
A certain question paper contains three parts A, B, C with four questions in each
part. It is required to answer seven questions selecting at least two questions from
10 2 1 L1
each part. In how many different ways a student select his seven questions for
answering
A women has 11 close relatives and she wishes to invite 5 of them to dinner. In how
many ways can she invite them in following situations i) there is no restriction on
11 2 1 L1
the choice, ii) two particular persons will not attend separately, iii) two particular
persons will not attend together.
How many arrangement are there for all the letters in the word
“SOCI0LOGICAL”.In how many of these arrangements,
12 2 1 L1
I) A and G are adjacent
II) All vowels are adjacent
Find the number of arrangements of all the letters in TALLAHASSE. In how many
13 2 1 L1
of these arrangements have no adjacent A’s
 14 Find the coefficients of
i) i) �9 �3 in the expansion of 2� − 3� 12
. 2 1 L1
2 15
ii) ii) �0 (Constant term) in the expansion of 3�2 − �
iii) iii) �11 �4 �2 in the expansion of 2�3 − 3��2 + �2 6 .
iv) iv) �2 �3 �2 �5 in the expansion of � + 2� − 3� + 2� + 5 16
In how many ways can one distribute eight identical balls into four distinct
15 containers so that i) no container is left empty 2 1 L1
ii) the four container gets an odd number of balls
In how many ways can one distribute 7 apples and 6 oranges among 4 children so
16 2 1 L1
that each child gets at least 1 apple
Determine the number of integer solutions of the equation
17 2 1 L1
�1 + �2 + �3 + �4 = 32, where �1 , �2 ≥ 5 : �3 , �4 ≥ 7 .
*CO- Course Outcome: Demonstrate the application of discrete structures in different fields of computer science.

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