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1
Kenneth Rosen Edition 7th Exercise 8.1 Question 15 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive $0s$ or two consecutive $1s.$ What are the initial conditions? How many ternary strings of length six do not contain two consecutive $0s$ or two consecutive $1s?$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

6
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Kenneth Rosen Edition 7th Exercise 8.1 Question 14 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length n that contain two consecutive $0s.$ What are the initial conditions? How many ternary strings of length six contain two consecutive $0s?$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

8
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Kenneth Rosen Edition 7th Exercise 8.1 Question 13 (Page No. 511)
A string that contains only $0s, 1s,$ and $2s$ is called a ternary string. Find a recurrence relation for the number of ternary strings of length $n$ that do not contain two consecutive $0s.$ What are the initial conditions? How many ternary strings of length six do not contain two consecutive $0s?$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

7
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Kenneth Rosen Edition 7th Exercise 8.1 Question 12 (Page No. 511)
Find a recurrence relation for the number of ways to climb $n$ stairs if the person climbing the stairs can take one, two, or three stairs at a time. What are the initial conditions? In many ways can this person climb a flight of eight stairs?
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

7
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Kenneth Rosen Edition 7th Exercise 8.1 Question 11 (Page No. 511)
Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time. What are the initial conditions? In how many ways can this person climb a flight of eight stairs?
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

7
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6
Kenneth Rosen Edition 7th Exercise 8.1 Question 9 (Page No. 511)
Find a recurrence relation for the number of bit strings of length n that do not contain three consecutive $0s.$ What are the initial conditions? How many bit strings of length seven do not contain three consecutive $0s?$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

9
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7
Kenneth Rosen Edition 7th Exercise 8.1 Question 8 (Page No. 511)
Find a recurrence relation for the number of bit strings of length $n$ that contain three consecutive $0s.$ What are the initial conditions? How many bit strings of length seven contain three consecutive $0s?$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

6
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8
Kenneth Rosen Edition 7th Exercise 8.1 Question 7 (Page No. 510  511)
Find a recurrence relation for the number of bit strings of length $n$ that contain a pair of consecutive $0s$. What are the initial conditions? How many bit strings of length seven contain two consecutive $0s?$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

8
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9
Kenneth Rosen Edition 7th Exercise 8.1 Question 6 (Page No. 510)
Find a recurrence relation for the number of strictly increasing sequences of positive integers that have 1 as their first term and n as their last term, where n is a positive integer. That is, sequences $a_{1}, a_{2},\dots,a_{k},$ ... How many sequences of the type described in $(A)$ are there when $n$ is an integer with $n \geq 2?$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

6
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10
Kenneth Rosen Edition 7th Exercise 8.1 Question 5 (Page No. 510)
How many ways are there to pay a bill of $17$ pesos using the currency described in question $4,$ where the order in which coins and bills are paid matters?
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

7
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11
Kenneth Rosen Edition 7th Exercise 8.1 Question 4 (Page No. 510)
A country uses as currency coins with values of $1$ peso, $2$ pesos, $5$ pesos, and $10$ pesos and bills with values of $5$ pesos, $10$ pesos, $20$ pesos, $50$ pesos, and $100$ pesos. Find a recurrence relation for the number of ways to pay a bill of $n$ pesos if the order in which the coins and bills are paid matters.
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

9
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12
Kenneth Rosen Edition 7th Exercise 8.1 Question 3 (Page No. 510)
A vending machine dispensing books of stamps accepts only onedollar coins, $\$1$ bills, and $\$5$ bills. Find a recurrence relation for the number of ways to deposit $n$ dollars in the vending machine, where the order in ... $10$ for a book of stamps?
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

7
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13
Kenneth Rosen Edition 7th Exercise 8.1 Question 2 (Page No. 510)
Find a recurrence relation for the number of permutations of a set with $n$ elements. Use this recurrence relation to find the number of permutations of a set with $n$ elements using iteration
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

6
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14
Kenneth Rosen Edition 7th Exercise 8.1 Question 1 (Page No. 510)
Use mathematical induction to verify the formula derived in Example $2$ for the number of moves required to complete the Tower of Hanoi puzzle.
asked
May 2
in
Combinatory
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Lakshman Patel RJIT

6
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15
Kenneth Rosen Edition 7th Exercise 6.6 Question 16 (Page No. 439)
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ This algorithm is based on Cantor expansions of integers. Every nonnegative integer less than $n!$ ... between Cantor expansions and permutations as described in the preamble to question $14.$ $3$ $89$ $111$
asked
May 2
in
Combinatory
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Lakshman Patel RJIT

19
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16
Kenneth Rosen Edition 7th Exercise 6.6 Question 17 (Page No. 438)
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ ... permutations of a set of n elements based on the correspondence described in the preamble to question $14.$
asked
May 2
in
Combinatory
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Lakshman Patel RJIT

8
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17
Kenneth Rosen Edition 7th Exercise 6.6 Question 15 (Page No. 438)
Show that the correspondence described in the preamble is a bijection between the set of permutations of $\{1, 2, 3,\dots,n\}$ and the nonnegative integers less than $n!.$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

7
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18
Kenneth Rosen Edition 7th Exercise 6.6 Question 14 (Page No. 438)
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ This algorithm is based on Cantor expansions of integers. Every nonnegative integer less than $n!$ has a unique ... $a_{1}, a_{2},\dots,a_{n−1}$ that correspond to these permutations. $246531$ $12345$ $654321$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

8
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19
Kenneth Rosen Edition 7th Exercise 6.6 Question 11 (Page No. 438)
Show that Algorithm $3$ produces the next larger $r$combination in lexicographic order after a given $r$combination.
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

7
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20
Kenneth Rosen Edition 7th Exercise 6.6 Question 10 (Page No. 438)
Show that Algorithm $1$ produces the next larger permutation in lexicographic order.
asked
May 2
in
Combinatory
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Lakshman Patel RJIT

7
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21
Kenneth Rosen Edition 7th Exercise 6.6 Question 9 (Page No. 438)
Use Algorithm $3$ to list all the $3$combinations of $\{1, 2, 3, 4, 5\}.$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

8
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22
Kenneth Rosen Edition 7th Exercise 6.6 Question 8 (Page No. 438)
Use Algorithm $2$ to list all the subsets of the set $\{1, 2, 3, 4\}.$
asked
May 2
in
Combinatory
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Lakshman Patel RJIT

6
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23
Kenneth Rosen Edition 7th Exercise 6.6 Question 7 (Page No. 438)
Use Algorithm $1$ to generate the $24$ permutations of the first four positive integers in lexicographic order.
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

8
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24
Kenneth Rosen Edition 7th Exercise 6.5 Question 64 (Page No. 434)
Find the expansion of $(x + y + z)^{4}.$
asked
May 1
in
Combinatory
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Lakshman Patel RJIT

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25
Kenneth Rosen Edition 7th Exercise 6.5 Question 63 (Page No. 434)
Prove the Multinomial Theorem: If $n$ ... $C(n:n_{1},n_{2},\dots,n_{m}) = \dfrac{n!}{n_{1}!n_{2}!\dots n_{m}!}$ is a multinomial coefficient.
asked
May 1
in
Combinatory
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Lakshman Patel RJIT

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26
Kenneth Rosen Edition 7th Exercise 6.5 Question 62 (Page No. 434)
How many different terms are there in the expansion of $(x_{1} + x_{2} +\dots + x_{m})^{n}$ after all terms with identical sets of exponents are added?
asked
May 1
in
Combinatory
by
Lakshman Patel RJIT

7
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27
Kenneth Rosen Edition 7th Exercise 6.5 Question 61 (Page No. 434)
Suppose that a weapons inspector must inspect each of five different sites twice, visiting one site per day. The inspector is free to select the order in which to visit these sites, but cannot visit site $\text{X},$ the most suspicious site, on two consecutive days. In how many different orders can the inspector visit these sites?
asked
May 1
in
Combinatory
by
Lakshman Patel RJIT

4
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Kenneth Rosen Edition 7th Exercise 6.5 Question 60 (Page No. 434)
Suppose that a basketball league has $32$ teams, split into two conferences of $16$ teams each. Each conference is split into three divisions. Suppose that the North Central Division has five teams. Each of the teams in the ... In how many different orders can the games of one of the teams in the North Central Division be scheduled?
asked
May 1
in
Combinatory
by
Lakshman Patel RJIT

6
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29
Kenneth Rosen Edition 7th Exercise 6.5 Question 59 (Page No. 434)
How many ways are there to distribute five balls into three boxes if each box must have at least one ball in it if both the balls and boxes are labeled? the balls are labeled, but the boxes are unlabeled? the balls are unlabeled, but the boxes are labeled? both the balls and boxes are unlabeled?
asked
May 1
in
Combinatory
by
Lakshman Patel RJIT

9
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Kenneth Rosen Edition 7th Exercise 6.5 Question 58 (Page No. 434)
How many ways are there to distribute five balls into seven boxes if each box must have at most one ball in it if both the balls and boxes are labeled? the balls are labeled, but the boxes are unlabeled? the balls are unlabeled, but the boxes are labeled? both the balls and boxes are unlabeled?
asked
May 1
in
Combinatory
by
Lakshman Patel RJIT

7
views
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