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Recent questions tagged discrete-mathematics
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331
Kenneth Rosen Edition 7 Exercise 6.6 Question 10 (Page No. 438)
Show that Algorithm $1$ produces the next larger permutation in lexicographic order.
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Combinatory
May 1, 2020
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admin
167
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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332
Kenneth Rosen Edition 7 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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Combinatory
May 1, 2020
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admin
179
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kenneth-rosen
discrete-mathematics
counting
combinatory
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Kenneth Rosen Edition 7 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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Combinatory
May 1, 2020
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admin
194
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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Kenneth Rosen Edition 7 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
May 1, 2020
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admin
267
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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Kenneth Rosen Edition 7 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
May 1, 2020
by
admin
782
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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Kenneth Rosen Edition 7 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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in
Combinatory
May 1, 2020
by
admin
1.6k
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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337
Kenneth Rosen Edition 7 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
May 1, 2020
by
admin
657
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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Kenneth Rosen Edition 7 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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in
Combinatory
May 1, 2020
by
admin
614
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
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339
Kenneth Rosen Edition 7 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
May 1, 2020
by
admin
552
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
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340
Kenneth Rosen Edition 7 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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in
Combinatory
May 1, 2020
by
admin
613
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
votes
1
answer
341
Kenneth Rosen Edition 7 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
May 1, 2020
by
admin
286
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
votes
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answer
342
Kenneth Rosen Edition 7 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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asked
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Combinatory
May 1, 2020
by
admin
285
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kenneth-rosen
discrete-mathematics
counting
combinatory
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343
Kenneth Rosen Edition 7 Exercise 6.5 Question 64 (Page No. 434)
Find the expansion of $(x + y + z)^{4}.$
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Combinatory
May 1, 2020
by
admin
199
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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Kenneth Rosen Edition 7 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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Combinatory
May 1, 2020
by
admin
522
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kenneth-rosen
discrete-mathematics
counting
combinatory
proof
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345
Kenneth Rosen Edition 7 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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Combinatory
May 1, 2020
by
admin
193
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
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Kenneth Rosen Edition 7 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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Combinatory
May 1, 2020
by
admin
591
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
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347
Kenneth Rosen Edition 7 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
May 1, 2020
by
admin
541
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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348
Kenneth Rosen Edition 7 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
May 1, 2020
by
admin
1.3k
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
1
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349
Kenneth Rosen Edition 7 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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in
Combinatory
May 1, 2020
by
admin
1.5k
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
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350
Kenneth Rosen Edition 7 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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Combinatory
May 1, 2020
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admin
530
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
1
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351
Kenneth Rosen Edition 7 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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in
Combinatory
May 1, 2020
by
admin
440
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
votes
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352
Kenneth Rosen Edition 7 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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asked
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Combinatory
May 1, 2020
by
admin
231
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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353
Kenneth Rosen Edition 7 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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Combinatory
May 1, 2020
by
admin
204
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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354
Kenneth Rosen Edition 7 Exercise 6.5 Question 53 (Page No. 434)
How many ways are there to put six temporary employees into four identical offices so that there is at least one temporary employee in each of these four offices?
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asked
in
Combinatory
May 1, 2020
by
admin
189
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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355
Kenneth Rosen Edition 7 Exercise 6.5 Question 52 (Page No. 434)
How many ways are there to put five temporary employees into four identical offices?
admin
asked
in
Combinatory
May 1, 2020
by
admin
261
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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356
Kenneth Rosen Edition 7 Exercise 6.5 Question 51 (Page No. 434)
How many ways are there to distribute six distinguishable objects into four indistinguishable boxes so that each of the boxes contains at least one object?
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in
Combinatory
May 1, 2020
by
admin
198
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
0
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Kenneth Rosen Edition 7 Exercise 6.5 Question 50 (Page No. 434)
How many ways are there to distribute five distinguishable objects into three indistinguishable boxes?
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in
Combinatory
May 1, 2020
by
admin
202
views
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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358
Kenneth Rosen Edition 7 Exercise 6.5 Question 49 (Page No. 433 - 434)
In this exercise we will prove Theorem $2$ by setting up a one-to-one correspondence between the set of $r$-combinations with repetition allowed of $S = \{1, 2, 3,\dots,n\}$ and the set of $r$ ... that there are $C(n + r − 1,r)\:\: r$-combinations with repetition allowed from a set with n elements.
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Combinatory
May 1, 2020
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admin
292
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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Kenneth Rosen Edition 7 Exercise 6.5 Question 48 (Page No. 433)
Prove Theorem $4$ by first setting up a one-to-one correspondence between permutations of $n$ objects with $n_{i}$ indistinguishable objects of type $i,\; i = 1, 2, 3,\dots, k,$ and the distributions of $n$ objects in $k$ boxes such that $n_{i}$ objects are placed in box $i,\: i = 1, 2, 3,\dots,k $ and then applying Theorem $3.$
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Combinatory
May 1, 2020
by
admin
239
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kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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360
Kenneth Rosen Edition 7 Exercise 6.5 Question 47 (Page No. 433)
Use the product rule to prove Theorem $4,$ by first placing objects in the first box, then placing objects in the second box, and so on.
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Combinatory
May 1, 2020
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admin
215
views
kenneth-rosen
discrete-mathematics
counting
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