Introductory Combinatorics
5th Edition
ISBN: 9780134689616
Author: Brualdi, Richard A.
Publisher: Pearson,
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Chapter 2, Problem 7E
To determine
The number of ways in which 4 men and 8 women be seated at a round table.
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What is the area of triangle ABC?
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D
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A 6√√3 square units
B 18√3 square units
36√3 square units
D 72√3 square units
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1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where v₁,..., Un are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1,2,-2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show…
Chapter 2 Solutions
Introductory Combinatorics
Ch. 2 - Prob. 1ECh. 2 - How many orderings are there for a deck of 52...Ch. 2 - In how many ways can a poker hand (five cards) be...Ch. 2 - How many distinct positive divisors does each of...Ch. 2 - Determine the largest power of 10 that is a factor...Ch. 2 - How many integers greater than 5400 have both of...Ch. 2 - In how many ways can four men and eight women be...Ch. 2 - In how many ways can six men and six women be...Ch. 2 - In how many ways can 15 people be seated at a...Ch. 2 - A committee of five people is to be chosen from a...
Ch. 2 - How many sets of three integers between 1 and 20...Ch. 2 - A football team of 11 players is to be selected...Ch. 2 - There are 100 students at a school and three...Ch. 2 - A classroom has two rows of eight seats each....Ch. 2 - At a party there are 15 men and 20 women.
How many...Ch. 2 - Prove that
by using a combinatorial argument and...Ch. 2 - In how many ways can six indistinguishable rooks...Ch. 2 - In how many ways can two red and four blue rooks...Ch. 2 - We are given eight rooks, five of which are red...Ch. 2 - Determine the number of circular permutations of...Ch. 2 - How many permutations are there of the letters of...Ch. 2 - A footrace takes place among four runners. If ties...Ch. 2 - Bridge is played with four players and an ordinary...Ch. 2 - Prob. 24ECh. 2 - A ferris wheel has five cars, each containing four...Ch. 2 - A group of mn people are to be arranged into m...Ch. 2 - In how many ways can five indistinguishable rooks...Ch. 2 - A secretary works in a building located nine...Ch. 2 - Prob. 29ECh. 2 - We are to seat five boys, five girls, and one...Ch. 2 - Prob. 31ECh. 2 - Determine the number of 11-permutations of the...Ch. 2 - Determine the number of 10-permutations of the...Ch. 2 - Determine the number of 11-permutations of the...Ch. 2 - List all 3-combintions and 4-combinations of the...Ch. 2 - Prob. 36ECh. 2 - A bakery sells six different kinds of pastry. If...Ch. 2 - How many integral solutions of
x1 + x2 + x3 + x4 =...Ch. 2 - There are 20 identical sticks lined up in a row...Ch. 2 - There are n sticks lined up in a row, and k of...Ch. 2 - In how many ways can 12 indistinguishable apples...Ch. 2 - Prob. 42ECh. 2 - Prob. 43ECh. 2 - Prove that the number of ways to distribute n...Ch. 2 - Prob. 45ECh. 2 - Prob. 46ECh. 2 - There are 2n + 1 identical books to be put in a...Ch. 2 - Prob. 48ECh. 2 - Prob. 49ECh. 2 - In how many ways can five identical rooks be...Ch. 2 - Consider the multiset {n · a, 1, 2, 3, … , n} of...Ch. 2 - Consider the multiset {n · a, n · b, 1, 2, 3, … ,...Ch. 2 - Find a one-to-one correspondence between the...Ch. 2 - Prob. 54ECh. 2 - How many permutations are there of the letters in...Ch. 2 - What is the probability that a poker hand contains...Ch. 2 - What is the probability that a poker hand contains...Ch. 2 - Prob. 58ECh. 2 - Prob. 59ECh. 2 - A bagel store sells six different kinds of bagels....Ch. 2 - Consider an 9-by-9 board and nine rooks of which...Ch. 2 - Prob. 62ECh. 2 - Four (standard) dice (cubes with 1, 2, 3, 4, 5, 6,...Ch. 2 - Let n be a positive integer. Suppose we choose a...
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- pleasd dont use chat gptarrow_forward1. True or false: (a) if E is a subspace of V, then dim(E) + dim(E+) = dim(V) (b) Let {i, n} be a basis of the vector space V, where vi,..., are all eigen- vectors for both the matrix A and the matrix B. Then, any eigenvector of A is an eigenvector of B. Justify. 2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1, 2, -2), (1, −1, 4), (2, 1, 1)}. 3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal projection onto the orthogonal complement E. (a) The combinations of projections P+Q and PQ correspond to well-known oper- ators. What are they? Justify your answer. (b) Show that P - Q is its own inverse. 4. Show that the Frobenius product on n x n-matrices, (A, B) = = Tr(B*A), is an inner product, where B* denotes the Hermitian adjoint of B. 5. Show that if A and B are two n x n-matrices for which {1,..., n} is a basis of eigen- vectors (for both A and B), then AB = BA. Remark: It is also true that if AB = BA, then there exists a common…arrow_forwardQuestion 1. Let f: XY and g: Y Z be two functions. Prove that (1) if go f is injective, then f is injective; (2) if go f is surjective, then g is surjective. Question 2. Prove or disprove: (1) The set X = {k € Z} is countable. (2) The set X = {k EZ,nЄN} is countable. (3) The set X = R\Q = {x ER2 countable. Q} (the set of all irrational numbers) is (4) The set X = {p.√2pQ} is countable. (5) The interval X = [0,1] is countable. Question 3. Let X = {f|f: N→ N}, the set of all functions from N to N. Prove that X is uncountable. Extra practice (not to be submitted). Question. Prove the following by induction. (1) For any nЄN, 1+3+5++2n-1 n². (2) For any nЄ N, 1+2+3++ n = n(n+1). Question. Write explicitly a function f: Nx N N which is bijective.arrow_forward
- 3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal projection onto the orthogonal complement E. (a) The combinations of projections P+Q and PQ correspond to well-known oper- ators. What are they? Justify your answer. (b) Show that P - Q is its own inverse.arrow_forwardAre natural logarithms used in real life ? How ? Can u give me two or three ways we can use them. Thanksarrow_forwardBy using the numbers -5;-3,-0,1;6 and 8 once, find 30arrow_forward
- Show that the Laplace equation in Cartesian coordinates: J²u J²u + = 0 მx2 Jy2 can be reduced to the following form in cylindrical polar coordinates: 湯( ди 1 8²u + Or 7,2 მ)2 = 0.arrow_forwardDraw the following graph on the interval πT 5π < x < x≤ 2 2 y = 2 cos(3(x-77)) +3 6+ 5 4- 3 2 1 /2 -π/3 -π/6 Clear All Draw: /6 π/3 π/2 2/3 5/6 x 7/6 4/3 3/2 5/311/6 2 13/67/3 5 Question Help: Video Submit Question Jump to Answerarrow_forwardDetermine the moment about the origin O of the force F4i-3j+5k that acts at a Point A. Assume that the position vector of A is (a) r =2i+3j-4k, (b) r=-8i+6j-10k, (c) r=8i-6j+5karrow_forward
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