Hot questions in Discrete Mathematics / GATE Overflow for GATE CSE

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1721

Kenneth Rosen Edition 7 Exercise 8.2 Question 31 (Page No. 525)
Find all solutions of the recurrence relation $a_{n} = 5a_{n-1} - 6a_{n-2} + 2^{n}+ 3n.$ [Hint: Look for a particular solution of the form $qn2^{n} + p_{1}n + p_{2},$ where $q, p_{1}, \text{and}\: p_{2}$ are constants.]

Find all solutions of the recurrence relation $a_{n} = 5a_{n-1} - 6a_{n-2} + 2^{n}+ 3n.$ [Hint: Look for a particular solution of the form $qn2^{n} + p_{1}n + p_{2},$ whe...

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admin asked May 5, 2020

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1 answer

1722

Kenneth Rosen Edition 7 Exercise 6.5 Question 30 (Page No. 433)
How many different strings can be made from the letters in $\text{MISSISSIPPI},$ using all the letters?

How many different strings can be made from the letters in $\text{MISSISSIPPI},$ using all the letters?

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318 views

admin asked Apr 30, 2020

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1723

Kenneth Rosen Edition 7 Exercise 8.2 Question 8 (Page No. 524 - 525)
A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. Find a recurrence relation for $\{L_{n}\},$ ... if $100,000$ lobsters were caught in year $1\:\text{ and}\: 300,000$ were caught in year $2.$

A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two ...

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admin asked May 3, 2020

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1724

Kenneth Rosen Edition 7 Exercise 6.3 Question 6 (Page No. 413)
Find the value of each of these quantities. $C(5, 1)$ $C(5, 3)$ $C(8, 4)$ $C(8, 8)$ $C(8, 0)$ $C(12, 6)$

Find the value of each of these quantities.$C(5, 1)$$C(5, 3)$$C(8, 4)$$C(8, 8)$$C(8, 0)$$C(12, 6)$

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389 views

admin asked Apr 29, 2020

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1725

Kenneth Rosen Edition 7 Exercise 8.2 Question 25 (Page No. 525)
Determine values of the constants $A$ and $B$ such that $a_{n} = A{n} + B$ is a solution of recurrence relation $a_{n} = 2a_{n-1} + n + 5.$ Use Theorem $5$ to find all solutions of this recurrence relation. Find the solution of this recurrence relation with $a_{0} = 4.$

Determine values of the constants $A$ and $B$ such that $a_{n} = A{n} + B$ is a solution of recurrence relation $a_{n} = 2a_{n-1} + n + 5.$Use Theorem $5$ to find all sol...

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300 views

admin asked May 5, 2020

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2 answers

1726

CMI2016-B-2a
A $\textit{simple path}$ (respectively cycle) in a graph is a path (respectively cycle) in which no edge or vertex is repeated. The $\textit{length}$ of such a path (respectively cycle) is the number of edges in the path (respectively cycle). Let $G$ be an undirected graph with minimum degree $k \geq 2$. Show that $G$ contains a simple path of length at least $k$.

A $\textit{simple path}$ (respectively cycle) in a graph is a path (respectively cycle) in which no edge or vertex is repeated. The $\textit{length}$ of such a path (resp...

go_editor

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go_editor asked Dec 30, 2016

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1 answer

1727

Kenneth Rosen Edition 7 Exercise 2.4 Question 3 (Page No. 167)
What are the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ of the sequence $\{a_{n}\},$ where $a_{n}$ equals $2^{n} + 1$ $(n + 1)^{n+1}$ $\left \lfloor n/2\right \rfloor$ $\left \lfloor n/2\right \rfloor + \left \lceil n/2\right \rceil$

What are the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ of the sequence $\{a_{n}\},$ where $a_{n}$ equals$2^{n} + 1$$(n + 1)^{n+1}$$\left \lfloor n/2\right \rfloor$$\left \...

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admin asked Apr 19, 2020

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1728

Kenneth Rosen Edition 7 Exercise 8.2 Question 43 (Page No. 526)
Express the solution of the linear nonhomogenous recurrence relation $a_{n} = a_{n-1} + a_{n-2} + 1\:\text{for}\: n \geq 2 \:\text{where}\: a_{0} = 0\:\text{and}\: a_{1} = 1$ in terms of the Fibonacci numbers. [Hint: Let $b_{n} = a_{n + 1}$ and apply question $42$ to the sequence $b_{n}.]$

Express the solution of the linear nonhomogenous recurrence relation $a_{n} = a_{n-1} + a_{n-2} + 1\:\text{for}\: n \geq 2\:\text{where}\: a_{0} = 0\:\text{and}\: a_{1} =...

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admin asked May 6, 2020

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1729

Kenneth Rosen Edition 7 Exercise 6.2 Question 35 (Page No. 406)
There are $38$ different time periods during which classes at a university can be scheduled. If there are $677$ different classes, how many different rooms will be needed?

There are $38$ different time periods during which classes at a university can be scheduled. If there are $677$ different classes, how many different rooms will be needed...

admin

499 views

admin asked Apr 29, 2020

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1730

Kenneth Rosen Edition 7 Exercise 6.5 Question 29 (Page No. 433)
How many different bit strings can be transmitted if the string must begin with a $1$ bit, must include three additional $1$ bits (so that a total of four $1$ bits is sent), must include a total of $12\:\: 0$ bits, and must have at least two $0$ bits following each $1$ bit?

How many different bit strings can be transmitted if the string must begin with a $1$ bit, must include three additional $1$ bits (so that a total of four $1$ bits is sen...

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385 views

admin asked Apr 30, 2020

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1 answer

1731

Kenneth Rosen Edition 7 Exercise 6.3 Question 43 (Page No. 415)
How many ways are there for a horse race with three horses to finish if ties are possible? [Note: Two or three horses may tie.]

How many ways are there for a horse race with three horses to finish if ties are possible? [Note: Two or three horses may tie.]

admin

379 views

admin asked Apr 29, 2020

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1732

Kenneth Rosen Edition 7 Exercise 8.2 Question 28 (Page No. 525)
Find all solutions of the recurrence relation $a_{n} = 2a_{n-1} + 2n^{2}.$ Find the solution of the recurrence relation in part $(A)$ with initial condition $a_{1} = 4.$

Find all solutions of the recurrence relation $a_{n} = 2a_{n-1} + 2n^{2}.$Find the solution of the recurrence relation in part $(A)$ with initial condition $a_{1} = 4.$

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206 views

admin asked May 5, 2020

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1733

Kenneth Rosen Edition 7 Exercise 8.2 Question 24 (Page No. 525)
Consider the nonhomogeneous linear recurrence relation $a_{n} = 2a_{n-1} + 2^{n}.$ Show that $a_{n} = n2^{n}$ is a solution of this recurrence relation. Use Theorem $5$ to find all solutions of this recurrence relation. Find the solution with $a_{0} = 2.$

Consider the nonhomogeneous linear recurrence relation $a_{n} = 2a_{n-1} + 2^{n}.$Show that $a_{n} = n2^{n}$ is a solution of this recurrence relation.Use Theorem $5$ to ...

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admin asked May 5, 2020

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1 answer

1734

Kenneth Rosen Edition 7 Exercise 6.5 Question 3 (Page No. 432)
How many strings of six letters are there?

How many strings of six letters are there?

admin

347 views

admin asked Apr 30, 2020

3 votes

1 answer

1735

TIFR CSE 2020 | Part B | Question: 13
Let $G$ be an undirected graph. An Eulerian cycle of $G$ is a cycle that traverses each edge of $G$ exactly once. A Hamiltonian cycle of $G$ is a cycle that traverses each vertex of $G$ exactly once. Which of the following ... has a Hamiltonian cycle A complete graph always has both an Eulerian cycle and a Hamiltonian cycle All of the other statements are true

Let $G$ be an undirected graph. An Eulerian cycle of $G$ is a cycle that traverses each edge of $G$ exactly once. A Hamiltonian cycle of $G$ is a cycle that traverses ea...

admin

817 views

admin asked Feb 10, 2020

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1 answer

1736

Kenneth Rosen Edition 7 Exercise 6.2 Question 12 (Page No. 405)
How many ordered pairs of integers $(a, b)$ are needed to guarantee that there are two ordered pairs $(a_{1}, b_{1})\: \text{and}\: (a_{2}, b_{2})$ such that $a_{1} \mod 5 = a_{2} \mod 5\:\text{and}\: b_{1} \mod 5 = b_{2} \mod 5?$

How many ordered pairs of integers $(a, b)$ are needed to guarantee that there are two ordered pairs $(a_{1}, b_{1})\: \text{and}\: (a_{2}, b_{2})$ such that $a_{1} \mod ...

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447 views

admin asked Apr 29, 2020

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1 answer

1737

Kenneth Rosen Edition 7 Exercise 6.3 Question 44 (Page No. 415)
How many ways are there for a horse race with four horses to finish if ties are possible? [Note: Any number of the four horses may tie.)

How many ways are there for a horse race with four horses to finish if ties are possible? [Note: Any number of the four horses may tie.)

admin

342 views

admin asked Apr 29, 2020

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1738

Kenneth Rosen Edition 7 Exercise 8.2 Question 36 (Page No. 526)
Let an be the sum of the first $n$ perfect squares, that is, $a_{n} = \displaystyle{}\sum_{k = 1}^{n} k^{2}.$ Show that the sequence $\{a_{n}\}$ ... initial condition $a_{1} = 1.$ Use Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.

Let an be the sum of the first $n$ perfect squares, that is, $a_{n} = \displaystyle{}\sum_{k = 1}^{n} k^{2}.$ Show that the sequence $\{a_{n}\}$ satisfies the linear nonh...

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195 views

admin asked May 5, 2020

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1739

Kenneth Rosen Edition 7 Exercise 8.2 Question 10 (Page No. 525)
Prove Theorem $2:$ Let $c_{1}$ and $c_{2}$ be real numbers with $c_{2}\neq 0.$ Suppose that $r^{2}-c_{1}r-c_{2} = 0$ has only one root $r_{0}.$ A sequence $\{a_{n}\}$ ... $n = 0,1,2,\dots,$ where $\alpha_{1}$ and $\alpha_{2}$ are constants.

Prove Theorem $2:$ Let $c_{1}$ and $c_{2}$ be real numbers with $c_{2}\neq 0.$ Suppose that $r^{2}-c_{1}r-c_{2} = 0$ has only one root $r_{0}.$ A sequence $\{a_{n}\}$ is ...

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242 views

admin asked May 3, 2020

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1740

Kenneth Rosen Edition 7 Exercise 6.4 Question 19 (Page No. 421)
Prove Pascal’s identity, using the formula for $\binom{n}{r}.$

Prove Pascal’s identity, using the formula for $\binom{n}{r}.$

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265 views

admin asked Apr 30, 2020

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