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Recent questions tagged kennethrosen
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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

7
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kennethrosen
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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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kennethrosen
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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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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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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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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
by
Lakshman Patel RJIT

6
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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
by
Lakshman Patel RJIT

19
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kennethrosen
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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
by
Lakshman Patel RJIT

8
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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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kennethrosen
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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

10
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Kenneth Rosen Edition 7th Exercise 6.6 Question 13 (Page No. 438)
List all $3$permutations of $\{1, 2, 3, 4, 5\}.$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

12
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Kenneth Rosen Edition 7th Exercise 6.6 Question 12 (Page No. 438)
Develop an algorithm for generating the $r$permutations of a set of $n$ elements.
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

9
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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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kennethrosen
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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
by
Lakshman Patel RJIT

7
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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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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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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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Kenneth Rosen Edition 7th Exercise 6.6 Question 6 (Page No. 438)
. Find the next larger permutation in lexicographic order after each of these permutations. $1342$ $45321$ $13245$ $612345$ $1623547$ f$23587416$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

12
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Kenneth Rosen Edition 7th Exercise 6.6 Question 5 (Page No. 438)
Find the next larger permutation in lexicographic order after each of these permutations. $1432$ $54123$ $12453$ $45231$ $6714235$ $31528764$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

14
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Kenneth Rosen Edition 7th Exercise 6.6 Question 4 (Page No. 438)
Suppose that the name of a file in a computer directory consists of three digits followed by two lowercase letters and each digit is $0, 1,\:\text{or}\: 2,$ and each letter is either $a\:\text{or}\: b.$ List the name of these files in lexicographic order, where we order letters using the usual alphabetic order of letters.
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

11
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Kenneth Rosen Edition 7th Exercise 6.6 Question 3 (Page No. 438)
The name of a file in a computer directory consists of three uppercase letters followed by a digit, where each letter is either $A, B,\:\text{ or}\: C,$ and each digit is either $1\: \text{or}\: 2.$ List the name of these files in lexicographic order, where we order letters using the usual alphabetic order of letters.
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

10
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 6.6 Question 2 (Page No. 438)
Place these permutations of $\{1,2,3,4,5,6\}$ in lexicographic order $:234561, 231456, 165432, 156423, 543216, 541236, 231465, 314562, 432561, 654321, 654312, 435612.$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

10
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Kenneth Rosen Edition 7th Exercise 6.6 Question 1 (Page No. 438)
Place these permutations of $\{1, 2, 3, 4, 5\}$ in lexicographic order $:43521, 15432, 45321, 23451, 23514, 14532, 21345, 45213, 31452, 31542.$
asked
May 2
in
Combinatory
by
Lakshman Patel RJIT

10
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Kenneth Rosen Edition 7th Exercise 6.5 Question 66 (Page No. 434)
How many terms are there in the expansion of $(x + y + z)^{100}?$
asked
May 1
in
Combinatory
by
Lakshman Patel RJIT

15
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Kenneth Rosen Edition 7th Exercise 6.5 Question 65 (Page No. 434)
Find the coefficient of $x^{3}y^{2}z^{5}\:\text{in}\: (x + y + z)^{10}.$
asked
May 1
in
Combinatory
by
Lakshman Patel RJIT

13
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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
by
Lakshman Patel RJIT

7
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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
by
Lakshman Patel RJIT

9
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kennethrosen
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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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kennethrosen
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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

5
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kennethrosen
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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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kennethrosen
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