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Most answered questions in Discrete Mathematics
58
votes
9
answers
31
GATE CSE 2017 Set 2 | Question: 47
If the ordinary generating function of a sequence $\left \{a_n\right \}_{n=0}^\infty$ is $\large \frac{1+z}{(1-z)^3}$, then $a_3-a_0$ is equal to ___________ .
If the ordinary generating function of a sequence $\left \{a_n\right \}_{n=0}^\infty$ is $\large \frac{1+z}{(1-z)^3}$, then $a_3-a_0$ is equal to ___________ .
Arjun
17.5k
views
Arjun
asked
Feb 14, 2017
Combinatory
gatecse-2017-set2
combinatory
generating-functions
numerical-answers
normal
+
–
25
votes
9
answers
32
GATE CSE 2017 Set 1 | Question: 47
The number of integers between $1$ and $500$ (both inclusive) that are divisible by $3$ or $5$ or $7$ is ____________ .
The number of integers between $1$ and $500$ (both inclusive) that are divisible by $3$ or $5$ or $7$ is ____________ .
Arjun
11.5k
views
Arjun
asked
Feb 14, 2017
Set Theory & Algebra
gatecse-2017-set1
set-theory&algebra
normal
numerical-answers
set-theory
+
–
23
votes
9
answers
33
GATE CSE 1987 | Question: 10e
Show that the conclusion $(r \to q)$ follows from the premises$:p, (p \to q) \vee (p \wedge (r \to q))$
Show that the conclusion $(r \to q)$ follows from the premises$:p, (p \to q) \vee (p \wedge (r \to q))$
makhdoom ghaya
5.0k
views
makhdoom ghaya
asked
Nov 14, 2016
Mathematical Logic
gate1987
mathematical-logic
propositional-logic
proof
descriptive
+
–
91
votes
9
answers
34
GATE CSE 2016 Set 1 | Question: 28
A function $f: \Bbb{N^+} \rightarrow \Bbb{N^+}$ , defined on the set of positive integers $\Bbb{N^+}$, satisfies the following properties: $f(n)=f(n/2)$ if $n$ is even $f(n)=f(n+5)$ if $n$ is odd Let $R=\{ i \mid \exists{j} : f(j)=i \}$ be the set of distinct values that $f$ takes. The maximum possible size of $R$ is ___________.
A function $f: \Bbb{N^+} \rightarrow \Bbb{N^+}$ , defined on the set of positive integers $\Bbb{N^+}$, satisfies the following properties: $f(n)=f(n/2)...
Sandeep Singh
21.3k
views
Sandeep Singh
asked
Feb 12, 2016
Set Theory & Algebra
gatecse-2016-set1
set-theory&algebra
functions
normal
numerical-answers
+
–
9
votes
9
answers
35
Kenneth Rosen Edition 6 Question 45 (Page No. 346)
How many bit strings of length eight contain either three consecutive 0s or four consecutive 1s?
How many bit strings of length eight contain either three consecutive 0s or four consecutive 1s?
Anu
8.9k
views
Anu
asked
Jul 13, 2015
Combinatory
combinatory
counting
+
–
32
votes
9
answers
36
GATE CSE 2015 Set 1 | Question: 54
Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is_______________.
Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is_______________.
makhdoom ghaya
24.4k
views
makhdoom ghaya
asked
Feb 13, 2015
Graph Theory
gatecse-2015-set1
graph-theory
graph-connectivity
normal
graph-planarity
numerical-answers
+
–
50
votes
9
answers
37
GATE IT 2005 | Question: 36
Let $P(x)$ and $Q(x)$ ...
Let $P(x)$ and $Q(x)$ be arbitrary predicates. Which of the following statements is always TRUE?$\left(\left(\forall x \left(P\left(x\right) \vee Q\left(x\right)\right)\r...
Ishrat Jahan
14.6k
views
Ishrat Jahan
asked
Nov 3, 2014
Mathematical Logic
gateit-2005
mathematical-logic
first-order-logic
normal
+
–
41
votes
9
answers
38
GATE IT 2004 | Question: 31
Let $p, q, r$ and $s$ be four primitive statements. Consider the following arguments: $P: [(¬p\vee q) ∧ (r → s) ∧ (p \vee r)] → (¬s → q)$ $Q: [(¬p ∧q) ∧ [q → (p → r)]] → ¬r$ $R: [[(q ∧ r) → p] ∧ (¬q \vee p)] → r$ $S: [p ∧ (p → r) ∧ (q \vee ¬ r)] → q$ Which of the above arguments are valid? $P$ and $Q$ only $P$ and $R$ only $P$ and $S$ only $P, Q, R$ and $S$
Let $p, q, r$ and $s$ be four primitive statements. Consider the following arguments:$P: [(¬p\vee q) ∧ (r → s) ∧ (p \vee r)] → (¬s → q)$$Q: [(¬p ∧q) �...
Ishrat Jahan
11.7k
views
Ishrat Jahan
asked
Nov 2, 2014
Mathematical Logic
gateit-2004
mathematical-logic
normal
propositional-logic
+
–
64
votes
9
answers
39
GATE IT 2006 | Question: 25
Consider the undirected graph $G$ defined as follows. The vertices of $G$ are bit strings of length $n$. We have an edge between vertex $u$ and vertex $v$ if and only if $u$ and $v$ differ in exactly one bit position (in other words, $v$ can be obtained from $u$ by ... $\left(\frac{1}{n}\right)$ $\left(\frac{2}{n}\right)$ $\left(\frac{3}{n}\right)$
Consider the undirected graph $G$ defined as follows. The vertices of $G$ are bit strings of length $n$. We have an edge between vertex $u$ and vertex $v$ if and only if ...
Ishrat Jahan
13.0k
views
Ishrat Jahan
asked
Oct 31, 2014
Graph Theory
gateit-2006
graph-theory
graph-coloring
normal
+
–
68
votes
9
answers
40
GATE IT 2008 | Question: 21
Which of the following first order formulae is logically valid? Here $\alpha(x)$ is a first order formula with $x$ as a free variable, and $\beta$ ... $[(\forall x, \alpha(x)) \rightarrow \beta] \rightarrow [\forall x, \alpha(x) \rightarrow \beta]$
Which of the following first order formulae is logically valid? Here $\alpha(x)$ is a first order formula with $x$ as a free variable, and $\beta$ is a first order formul...
Ishrat Jahan
14.8k
views
Ishrat Jahan
asked
Oct 27, 2014
Mathematical Logic
gateit-2008
first-order-logic
normal
+
–
41
votes
9
answers
41
GATE CSE 1996 | Question: 2.1
Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, x-y)$. The inverse function of $f$ is given by $f^{-1} (x,y) = \left( \frac {1}{x+y}, \frac{1}{x-y}\right)$ ... $f^{-1}(x,y)=\left [ 2\left(x-y\right),2\left(x+y\right) \right ]$
Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, x-y)$. The inverse function of $f$ is ...
Kathleen
9.6k
views
Kathleen
asked
Oct 9, 2014
Set Theory & Algebra
gate1996
set-theory&algebra
functions
normal
+
–
38
votes
9
answers
42
GATE CSE 2014 Set 2 | Question: 3
The maximum number of edges in a bipartite graph on $12$ vertices is____
The maximum number of edges in a bipartite graph on $12$ vertices is____
go_editor
26.9k
views
go_editor
asked
Sep 28, 2014
Graph Theory
gatecse-2014-set2
graph-theory
graph-connectivity
numerical-answers
normal
+
–
60
votes
9
answers
43
GATE CSE 2005 | Question: 44
What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $(a,b)$ and $(c,d)$ in the chosen set such that, $a \equiv c\mod 3$ and $b \equiv d \mod 5$ $4$ $6$ $16$ $24$
What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $(a,b)$ and $(c,d)$ in the chosen set such th...
gatecse
13.3k
views
gatecse
asked
Sep 21, 2014
Combinatory
gatecse-2005
set-theory&algebra
normal
pigeonhole-principle
+
–
65
votes
9
answers
44
GATE CSE 2004 | Question: 75
Mala has the colouring book in which each English letter is drawn two times. She wants to paint each of these $52$ prints with one of $k$ colours, such that the colour pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $k$ that satisfies this requirement? $9$ $8$ $7$ $6$
Mala has the colouring book in which each English letter is drawn two times. She wants to paint each of these $52$ prints with one of $k$ colours, such that the colour pa...
Kathleen
16.5k
views
Kathleen
asked
Sep 18, 2014
Combinatory
gatecse-2004
combinatory
+
–
76
votes
9
answers
45
GATE CSE 2006 | Question: 25
Let $S = \{1, 2, 3,\ldots, m\}, m >3.$ Let $X_1,\ldots,X_n$ be subsets of $S$ each of size $3.$ Define a function $f$ from $S$ to the set of natural numbers as, $f(i)$ is the number of sets $X_j$ that contain the element $i.$ That is $f(i)=\left | \left\{j \mid i\in X_j \right\} \right|$ then $ \sum_{i=1}^{m} f(i)$ is: $3m$ $3n$ $2m+1$ $2n+1$
Let $S = \{1, 2, 3,\ldots, m\}, m >3.$ Let $X_1,\ldots,X_n$ be subsets of $S$ each of size $3.$ Define a function $f$ from $S$ to the set of natural numbers as, $f(i)$ is...
Rucha Shelke
10.9k
views
Rucha Shelke
asked
Sep 18, 2014
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
functions
+
–
65
votes
9
answers
46
GATE CSE 2003 | Question: 40
A graph $G=(V,E)$ satisfies $\mid E \mid \leq 3 \mid V \mid - 6$. The min-degree of $G$ is defined as $\min_{v\in V}\left\{ \text{degree }(v)\right \}$. Therefore, min-degree of $G$ cannot be $3$ $4$ $5$ $6$
A graph $G=(V,E)$ satisfies $\mid E \mid \leq 3 \mid V \mid - 6$. The min-degree of $G$ is defined as $\min_{v\in V}\left\{ \text{degree }(v)\right \}$. Therefore, min-d...
Kathleen
15.5k
views
Kathleen
asked
Sep 17, 2014
Graph Theory
gatecse-2003
graph-theory
normal
degree-of-graph
+
–
60
votes
9
answers
47
GATE CSE 2003 | Question: 36
How many perfect matching are there in a complete graph of $6$ vertices? $15$ $24$ $30$ $60$
How many perfect matching are there in a complete graph of $6$ vertices?$15$$24$$30$$60$
Kathleen
45.1k
views
Kathleen
asked
Sep 16, 2014
Graph Theory
gatecse-2003
graph-theory
graph-matching
normal
+
–
23
votes
9
answers
48
GATE CSE 2003 | Question: 34
$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be placed in the bags if each bag must contain at least $k$ ... $\left( \begin{array}{c} m - kn + n + k - 2 \\ n - k \end{array} \right)$
$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be place...
Kathleen
11.1k
views
Kathleen
asked
Sep 16, 2014
Combinatory
gatecse-2003
combinatory
balls-in-bins
normal
+
–
40
votes
9
answers
49
GATE CSE 2009 | Question: 22
For the composition table of a cyclic group shown below: ... $a,b$ are generators $b,c$ are generators $c,d$ are generators $d,a$ are generators
For the composition table of a cyclic group shown below:$$\begin{array}{|c|c|c|c|c|} \hline \textbf{*} & \textbf{a}& \textbf{b} &\textbf{c} & \textbf{d}\\\hline \textbf{a...
gatecse
8.7k
views
gatecse
asked
Sep 15, 2014
Set Theory & Algebra
gatecse-2009
set-theory&algebra
normal
group-theory
+
–
38
votes
9
answers
50
GATE CSE 2001 | Question: 2.1
How many $4$-digit even numbers have all $4$ digits distinct? $2240$ $2296$ $2620$ $4536$
How many $4$-digit even numbers have all $4$ digits distinct?$2240$$2296$$2620$$4536$
Kathleen
12.5k
views
Kathleen
asked
Sep 14, 2014
Combinatory
gatecse-2001
combinatory
normal
+
–
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