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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 $14.$ $3$ $89$ $111$
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Combinatory
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Lakshman Patel RJIT
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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 $14.$
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Combinatory
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Lakshman Patel RJIT
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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 $n!.$
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Combinatory
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Lakshman Patel RJIT
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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. $246531$ $12345$ $654321$
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Combinatory
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Lakshman Patel RJIT
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Kenneth Rosen Edition 7th Exercise 6.6 Question 13 (Page No. 438)
List all $3$permutations of $\{1, 2, 3, 4, 5\}.$
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Combinatory
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Lakshman Patel RJIT
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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.
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Combinatory
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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.
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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.
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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 $\{1, 2, 3, 4, 5\}.$
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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 $\{1, 2, 3, 4\}.$
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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.
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Combinatory
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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$23587416$
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Combinatory
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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. $1432$ $54123$ $12453$ $45231$ $6714235$ $31528764$
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Combinatory
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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.
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Combinatory
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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.
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Combinatory
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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 $:234561, 231456, 165432, 156423, 543216, 541236, 231465, 314562, 432561, 654321, 654312, 435612.$
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Combinatory
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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 $:43521, 15432, 45321, 23451, 23514, 14532, 21345, 45213, 31452, 31542.$
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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 $(x + y + z)^{100}?$
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Combinatory
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Kenneth Rosen Edition 7th Exercise 6.5 Question 65 (Page No. 434)
Find the coefficient of $x^{3}y^{2}z^{5}\:\text{in}\: (x + y + z)^{10}.$
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Kenneth Rosen Edition 7th Exercise 6.5 Question 64 (Page No. 434)
Find the expansion of $(x + y + z)^{4}.$
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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.
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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?
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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?
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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?
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Combinatory
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Lakshman Patel RJIT
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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?
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Combinatory
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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?
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Combinatory
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Kenneth Rosen Edition 7th Exercise 6.5 Question 57 (Page No. 434)
How many ways are there to pack nine identical DVDs into three indistinguishable boxes so that each box contains at least two DVDs?
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Kenneth Rosen Edition 7th Exercise 6.5 Question 56 (Page No. 434)
How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD?
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Kenneth Rosen Edition 7th Exercise 6.5 Question 55 (Page No. 434)
How many ways are there to distribute six indistinguishable objects into four indistinguishable boxes so that each of the boxes contains at least one object?
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Kenneth Rosen Edition 7th Exercise 6.5 Question 54 (Page No. 434)
How many ways are there to distribute five indistinguishable objects into three indistinguishable boxes?
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