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1
Kenneth Rosen Edition 7th Exercise 6.4 Question 27 (Page No. 422)
Prove the hockeystick identity $\displaystyle{}\sum_{k=0}^{r} \binom{n + k}{k} = \binom{n + r + 1}{r}$ whenever $n$ and $r$ are positive integers, using a combinatorial argument. using Pascal’s identity.
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
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0
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2
Kenneth Rosen Edition 7th Exercise 6.4 Question 26 (Page No. 422)
Let $n$ and $k$ be integers with $1 \leq k \leq n.$ Show that $\displaystyle{}\sum_{k=1}^{n} \binom{n}{k}\binom{n}{k − 1} = \dfrac{\binom{2n + 2}{n + 1}}{2} − \binom{2n}{n}.$
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

7
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
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3
Kenneth Rosen Edition 7th Exercise 6.4 Question 24 (Page No. 422)
Show that if $p$ is a prime and $k$ is an integer such that $1 \leq k \leq p − 1,$ then $p$ divides $\binom{p}{k} .$
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

8
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
votes
0
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4
Kenneth Rosen Edition 7th Exercise 6.4 Question 23 (Page No. 422)
Show that if $n$ and $k$ are positive integers, then $\binom{n + 1}{k} = \dfrac{(n + 1)\binom {n}{k – 1}}{k}.$ Use this identity to construct an inductive definition of the binomial coefficients.
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
votes
0
answers
5
Kenneth Rosen Edition 7th Exercise 6.4 Question 22 (Page No. 422)
Prove the identity $\binom{n}{r}\binom{r}{k} = \binom{n}{k}\binom{n−k}{r−k} ,$ whenever $n, r,$ and $k$ are nonnegative integers with $r \leq n$ and $k \leq r,$ using a combinatorial argument. using an argument based on the formula for the number of $r$combinations of a set with $n$ elements.
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

5
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
votes
0
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6
Kenneth Rosen Edition 7th Exercise 6.4 Question 21 (Page No. 422)
Prove that if $n$ and $k$ are integers with $1 \leq k \leq n,$ then $k \binom{n}{k} = n \binom{n−1}{k−1},$ using a combinatorial proof. [Hint: Show that the two sides of the identity count the number of ways to select ... subset.] using an algebraic proof based on the formula for $\binom{n}{r}$ given in Theorem $2$ in Section $6.3.$
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

5
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
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7
Kenneth Rosen Edition 7th Exercise 6.4 Question 20 (Page No. 421)
Suppose that $k$ and $n$ are integers with $1 \leq k<n.$ Prove the hexagon identity $\binom{n1}{k1}\binom{n}{k+1}\binom{n+1}{k} = \binom{n1}{k}\binom{n}{k1}\binom{n+1}{k+1},$ which relates terms in Pascal’s triangle that form a hexagon.
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

9
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kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
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8
Kenneth Rosen Edition 7th Exercise 6.4 Question 19 (Page No. 421)
Prove Pascal’s identity, using the formula for $\binom{n}{r}.$
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

5
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
votes
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9
Kenneth Rosen Edition 7th Exercise 6.4 Question 18 (Page No. 421)
Suppose that $b$ is an integer with $b \geq 7.$ Use the binomial theorem and the appropriate row of Pascal’s triangle to find the base$b$ expansion of $(11)^{4}_{b}$ [that is, the fourth power of the number $(11)_{b}$ in base$b$ notation].
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

12
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
votes
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10
Kenneth Rosen Edition 7th Exercise 6.4 Question 17 (Page No. 421)
Show that if $n$ and $k$ are integers with $1 \leq k \leq n,$ then $\binom{n}{k} \leq \frac{n^{k}}{2^{k−1}}.$
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
votes
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11
Kenneth Rosen Edition 7th Exercise 6.4 Question 16 (Page No. 421)
Use question $14$ and Corollary $1$ to show that if $n$ is an integer greater than $1,$ then $\binom{n}{\left \lfloor n/2 \right \rfloor}\geq \frac{2^{n}}{2}.$ Conclude from part $(A)$ that if $n$ is a positive integer, then $\binom{2n}{n}\geq \frac{4^{n}}{2n}.$
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

7
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kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
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12
Kenneth Rosen Edition 7th Exercise 6.4 Question 15 (Page No. 421)
Show that $\binom{n}{k} \leq 2^{n}$ for all positive integers $n$ and all integers $k$ with $0 \leq k \leq n.$
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

8
views
kennethrosen
discretemathematics
counting
binomialtheorem
descriptive
0
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13
Kenneth Rosen Edition 7th Exercise 6.3 Question 46 (Page No. 415)
This procedure is used to break ties in games in the championship round of the World Cup soccer tournament. Each team selects five players in a prescribed order. Each of these players takes a penalty kick, with a player from ... settled with no more than $10$ total additional kicks after the two rounds of five kicks for each team?
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

14
views
kennethrosen
discretemathematics
counting
combinatory
descriptive
0
votes
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14
Kenneth Rosen Edition 7th Exercise 6.3 Question 45 (Page No. 415)
There are six runners in the $100$yard dash. How many ways are there for three medals to be awarded if ties are possible? (The runner or runners who finish with the fastest time receive gold medals, the ... ahead receive silver medals, and the runner or runners who finish with exactly two runners ahead receive bronze medals.)
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

8
views
kennethrosen
discretemathematics
counting
combinatory
descriptive
0
votes
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15
Kenneth Rosen Edition 7th Exercise 6.3 Question 42 (Page No. 415)
Find a formula for the number of ways to seat $r$ of $n$ people around a circular table, where seatings are considered the same if every person has the same two neighbors without regard to which side these neighbors are sitting on.
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

5
views
kennethrosen
discretemathematics
counting
combinatory
descriptive
0
votes
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16
Kenneth Rosen Edition 7th Exercise 6.3 Question 41 (Page No. 415)
A circular $r$permutation of $n$ people is a seating of $r$ of these $n$ people around a circular table, where seatings are considered to be the same if they can be obtained from each other by rotating the table. Find a formula for the number of circular $r$permutations of $n$ people.
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
combinatory
descriptive
0
votes
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17
Kenneth Rosen Edition 7th Exercise 6.3 Question 40 (Page No. 415)
A circular $r$permutation of $n$ people is a seating of $r$ of these $n$ people around a circular table, where seatings are considered to be the same if they can be obtained from each other by rotating the table. Find the number of circular 3permutations of 5 people.
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

5
views
kennethrosen
discretemathematics
counting
combinatory
descriptive
0
votes
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18
Kenneth Rosen Edition 7th Exercise 6.3 Question 39 (Page No. 415)
How many license plates consisting of three letters followed by three digits contain no letter or digit twice?
asked
Apr 30
in
Combinatory
by
Lakshman Patel RJIT

8
views
kennethrosen
discretemathematics
counting
combinatory
descriptive
0
votes
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19
Kenneth Rosen Edition 7th Exercise 6.3 Question 38 (Page No. 414)
How many ways are there to select $12$ countries in the United Nations to serve on a council if $3$ are selected from a block of $45, 4$ are selected from a block of $57,$ and the others are selected from the remaining $69$ countries?
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

13
views
kennethrosen
discretemathematics
counting
combinatory
descriptive
0
votes
0
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20
Kenneth Rosen Edition 7th Exercise 6.2 Question 47 (Page No. 407)
An alternative proof of Theorem $3$ ... there is no increasing subsequence of length $n + 1,$ then there must be a decreasing subsequence of this length.
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
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0
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21
Kenneth Rosen Edition 7th Exercise 6.2 Question 46 (Page No. 407)
Let $n_{1}, n_{2},\dots,n_{t}$ be positive integers. Show that if $n_{1} + n_{2} +\dots + n_{t} − t + 1$ objects are placed into $t$ boxes, then for some $i, i = 1, 2,\dots,t,$ the $i^{\text{th}}$ box contains at least $n_{i}$ objects.
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

7
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
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0
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22
Kenneth Rosen Edition 7th Exercise 6.2 Question 45 (Page No. 407)
Let $x$ be an irrational number. Show that for some positive integer $j$ not exceeding the positive integer $n,$ the absolute value of the difference between $j x$ and the nearest integer to $j x$ is less than $1/n.$
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

5
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
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23
Kenneth Rosen Edition 7th Exercise 6.2 Question 44 (Page No. 406)
There are $51$ houses on a street. Each house has an address between $1000\: \text{and}\: 1099,$ inclusive. Show that at least two houses have addresses that are consecutive integers.
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
votes
0
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24
Kenneth Rosen Edition 7th Exercise 6.2 Question 43 (Page No. 406)
Show that if $f$ is a function from $S\: \text{to}\: T,$ where $S\: \text{and}\: T$ are nonempty finite sets and $m = \left \lceil \mid S \mid / \mid T \mid \right \rceil ,$ then there are at least $m$ elements of $S$ mapped to the same value ... $f (s_{1}) = f (s_{2}) =\dots = f (s_{m}).$
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

7
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
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25
Kenneth Rosen Edition 7th Exercise 6.2 Question 42 (Page No. 406)
Is the statement in question $41$ true if $24$ is replaced by $2?$ $23?$ $25?$ $30?$
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
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26
Kenneth Rosen Edition 7th Exercise 6.2 Question 41 (Page No. 406)
An arm wrestler is the champion for a period of $75$ hours. (Here, by an hour, we mean a period starting from an exact hour, such as $1\: \text{p.m.,}$ until the next hour.) The arm wrestler had at ... than $125$ total matches. Show that there is a period of consecutive hours during which the arm wrestler had exactly $24$ matches.
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

9
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
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27
Kenneth Rosen Edition 7th Exercise 6.2 Question 39 (Page No. 406)
Find the least number of cables required to connect $100$ computers to $20$ printers to guarantee that $2$ every subset of $20 $computers can directly access $20$ different printers. (Here, the assumptions about cables and computers are the same as in Example $9.$) Justify your answer.
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
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0
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28
Kenneth Rosen Edition 7th Exercise 6.2 Question 38 (Page No. 406)
Find the least number of cables required to connect eight computers to four printers to guarantee that for every choice of four of the eight computers, these four computers can directly access four different printers. Justify your answer.
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

8
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
votes
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29
Kenneth Rosen Edition 7th Exercise 6.2 Question 37 (Page No. 406)
A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of ... impossible to have a computer linked to none of the others and a computer linked to all the others.]
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

6
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
descriptive
0
votes
0
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30
Kenneth Rosen Edition 7th Exercise 6.2 Question 36 (Page No. 406)
A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers.
asked
Apr 29
in
Combinatory
by
Lakshman Patel RJIT

5
views
kennethrosen
discretemathematics
counting
pigeonholeprinciple
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