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linear algebra
Questions and Answers of
Linear Algebra
Let S be a finite set with |S| = N and let c1, c2, c3, c4 be four conditions, each of which may be satisfied by one or more of the elements of S. Prove that N(234) = N(c1234) + N(1234).
Professor Bailey has just completed writing the final examination for his course in advanced engineering mathematics. This examination has 12 questions, whose total value is to be 200 points. In how
At Flo's Flower Shop, Flo wants to arrange 15 different plants on five shelves for a window display. In how many ways can she arrange them so that each shelf has at least one, but no more than four,
In how many ways can Troy select nine marbles from a bag of twelve (identical except for color), where three are red, three blue, three white, and three green?
Find the number of permutations of a, b, c, . . . , x, y, z, in which none of the patterns spin, game, path, or net occurs.
Answer the question in Example 8.10 for the case of six villages. Example 8.10 In a certain area of the countryside are five villages. An engineer is to devise a system of two-way roads so that after
In how many ways can three x's, three y's, and three z's be arranged so that no consecutive triple of the same letter appears?
Frostburg township sponsors four Boy Scout troops, each with 20 boys. If the head scoutmaster selects 50 of these boys to represent this township at the state jamboree, what is the probability that
If Zachary rolls a fair die five times, what is the probability that the sum of his five rolls is 20?
Establish the Principle of Inclusion and Exclusion by applying the Principle of Mathematical Induction to the number t of conditions.
At a 12-week conference in mathematics, Sharon met seven of her friends from college. During the conference she met each friend at lunch 35 times, every pair of them 16 times, every trio eight times
Compute ϕ(n) for n equal to (a) 5186; (b) 5187; (c) 5188.
How many positive integers n less than 6000 (a) Satisfy gcd(n, 6000) = 1? (b) Share a common prime divisor with 6000?
If m, n ∈ Z+, prove that ϕ(nm) = nm-1ϕ(n).
For which positive integers n is ϕ(n) a power of 2?
For which positive integers n does 4 divide ϕ(n)?
Of the 100 students in Example 8.3, how many are taking (a) Fundamentals of Computer Programming but none of the other three courses; (b) Fundamentals of Computer Programming and Introduction to
At an upcoming family reunion, five families - each consisting of a husband, wife, and one child - are to be seated around a circular table. In how many ways can these 15 people be arranged around
Annually, the 65 members of the maintenance staff sponsor a "Christmas in July" picnic for the 400 summer employees at their company. For these 65 people, 21 bring hot dogs, 35 bring fried chicken,
Determine the number of positive integers n, 1 ≤ n ≤ 2000, that are (a) Not divisible by 2, 3, or 5 (b) Not divisible by 2, 3, 5, or 7 (c) Not divisible by 2, 3, or 5, but are divisible by 7
Determine how many integer solutions there are to x1 + x2 + x3 + x4 = 19, if (a) 0 ≤ x1 for all 1 ≤ i ≤ 4 (b) 0 ≤ x1 < 8 for all 1 ≤ i ≤ 4 (c) 0 ≤ x1 ≤ 5, 0 ≤ x2 ≤ 6, 3 ≤ x3 ≤
In how many ways can one arrange all of the letters in the word INFORMATION so that no pair of consecutive letters occurs more than once? [Here we want to count arrangements such as IINNOOFRMTA and
Determine the number of integer solutions to x1 + x2 + x3 + x4 = 19 where - 5 < x1 < 10 for all 1 ≤ i ≤ 4.
Determine the number of positive integers x where x ≤ 9,999,999 and the sum of the digits in x equals 31.
For the situation in Examples 8.10 and 8.11 compute Et for 0 ¤ i ¤ 5 and show that
(a) In how many ways can the letters in ARRANGEMENT be arranged so that there are exactly two pairs of consecutive identical letters? At least two pairs of consecutive identical letters?(b) Answer
In how many ways can one arrange the letters in CORRESPONDENTS so that (a) There is no pair of consecutive identical letters? (b) There are exactly two pairs of consecutive identical letters? (c)
Let A = {1, 2, 3,..., 10}, and B = {1,2,3, ..., 7}. How many functions f: A → B satisfy |f(A)| = 4? How many have |f(A)| ≤ 4?
In how many ways can one distribute ten distinct prizes among four students with exactly two students getting nothing? How many ways have at least two students getting nothing?
Zelma is having a luncheon for herself and nine of the women in her tennis league. On the morning of the luncheon she places name cards at the ten places at her table and then leaves to run a
If 13 cards are dealt from a standard deck of 52, what is the probability that these 13 cards include (a) At least one card from each suit? (b) Exactly one void (for example, no clubs)? (c) Exactly
The following provides an outline for proving Corollary 8.2. Fill in the needed details.(a) What is Et-1, and how are Lt and Lt-1 related?(b) Show that(c) For all 1 ¤ m ¤ t
In how many ways can the integers 1, 2, 3,..., 10 be arranged in a line so that no even integer is in its natural position?
(a) When n balls, numbered 1, 2, 3, ... , n are taken in succession from a container, a rencontre occurs if the mth ball withdrawn is numbered m, for some 1 ≤ m ≤ n. Find the probability of
Ten women attend a business luncheon. Each woman checks her coat and attache case. Upon leaving, each woman is given a coat and case at random. (a) In how many ways can the coats and cases be
Give a combinatorial argument to verify that for all n ˆˆ Z+,(For each 1 ‰¤ k ‰¤ n, dk = the number of derangements of 1, 2, 3, . . . , k; d0 = 1.)
(a) In how many ways can the integers 1, 2, 3, . . . , n be arranged in a line so that none of the patterns 12, 23, 34,..., (n - 1)n occurs? (b) Show that the result in part (a) equals dn-1 + dn. (dn
Answer part (a) of Exercise 14 if the numbers are arranged in a circle, and, as we count clockwise about the circle, none of the patterns 12, 23, 34, ..., (n - 1)n, n1 occurs. Exercise 14 In how many
(a) List all the derangements of 1, 2, 3, 4, 5 where the first three numbers are 1, 2, and 3, in some order. (b) List all the derangements of 1, 2, 3, 4, 5, 6 where the first three numbers are 1, 2,
How many derangements are there for 1, 2, 3, 4, 5?
How many permutations of 1, 2, 3, 4, 5, 6, 7 are not derangements?
A pair of dice, one red and the other green, is rolled six times. We know that the ordered pairs (1, 1), (1, 5), (2, 4), (3, 6), (4, 2), (4, 4), (5, 1), and (5, 5) did not come up. What is the
A computer dating service wants to match each of four women with one of six men. According to the information these applicants provided when they joined the service, we can draw the following
For A = {1, 2, 3, 4, 5} and B = {u, v, w, x, y, z}, determine the number of one-to-one functions f: A → B where f(1) ≠ V, w; f(2) ≠ u, w; f(3) ≠ x; and f(4) ≠ v, x, y.
(a) Find the rook polynomial for the standard 8 × 8 chessboard. (b) Answer part (a) with 8 replaced by n, for n ∈ Z+.
(a) Find the rook polynomials for the shaded chessboards in Fig. 8.13.(b) Generalize the chessboard (and rook polynomial) for Fig. 8.13(i).
(a) Let C be a chessboard that has m rows and n columns, with m ≤ n (for a total of mn squares). For 0 ≤ k ≤ m, in how many ways can we arrange k (identical) nontaking rooks on C?(b) For the
Professor Ruth has five graders to correct programs in her courses in Java, C++, SQL, Perl, and VHDL. Graders Jeanne and Charles both dislike SQL, Sandra wants to avoid C++ and VHDL. Paul detests
Determine how many n ∈ Z+ satisfy n ≤ 500 and are not divisible by 2, 3, 5, 6, 8, or 10.
In how many ways can four w's, four x's, four y's, and four z's be arranged so that there is no consecutive quadruple of the same letter?
(a) Given n distinct objects, in how many ways can we select r of these objects so that each selection includes some particular m of the n objects? (Here m ¤ r ¤ n.)(b)
(a) Let λ Z+. If we have λ different colors available, in how many ways can we color the vertices of the graph shown in Fig. 8.14(a) so that no adjacent
Find the number of ways to arrange the letters in LAPTOP so that none of the letters L, A, T, O is in its original position and the letter P is not in the third or sixth position.
For n ∈ Z+ prove that if ϕ(n) = n - 1 then n is prime.
Let D18 denote the set of positive divisors of 18. For d ∈ D18 let Sd = {n|0 < n ≤ 18 and gcd(n, 18) = d}. (a) Show that the collection Sd, d ∈ D18, provides a partition of {1, 2, 3, 4, ... ,
For m ∈ Z+ let Dm = {d ∈ Z+ | d divides m}. For d ∈ Dm let Sd = {n|0 < n ≤ m and gcd(n, m) = d}. (a) Show that the collection Sd, d ∈ Dm, provides a partition of {1, 2, 3, 4,..., m - 1,
If n ∈ Z+, prove that(a) ϕ(2n) = 2ϕ(n) when n is even; and(b) ϕ(2n) = ϕ(n) when n is odd.
Let a, b, c ∈ Z+ with c = gcd(a, b). Prove that ϕ(ab)ϕ(c) = ϕ(a)ϕ(b)c.
How many integers n are such that 0 ≤ n < 1,000,000 and the sum of the digits in n is less than or equal to 37?
At next week's church bazaar, Joseph and his cousin Jeffrey must arrange six baseballs, six footballs, six soccer balls, and six volleyballs on the four shelves in the sports booth sponsored by their
Find the number of positive integers n where 1 ≤ n ≤ 1000 and n is not a perfect square, cube, or fourth power.
In how many ways can we arrange the integers 1, 2, 3,..., 8 in a line so that there are no occurrences of the patterns 12, 23, ... , 78, 81?
(a) If we have k different colors available, in how many ways can we paint the walls of a pentagonal room if adjacent walls are to be painted with different colors? (b) What is the smallest value of
Ten students take a physics test in a certain room. When the test is over the students take a break and then return to the room to discuss their answers to the test questions. If there are 14 chairs
Using the result of Theorem 8.2, prove that the number of ways we can place s different objects in n distinct containers with m containers each containing exactly r of the objects is
If an arrangement of the letters in SURREPTITIOUS is selected at random, what is the probability that it contains (a) (Exactly) three pairs of consecutive identical letters? (b) At most three pairs
For each of the following, determine a generating function and indicate the coefficient in the function that is needed to solve the problem. (Give both the polynomial and power series forms of the
Determine the generating function for the number of ways to distribute 35 pennies (from an unlimited supply) among five children if (a) there are no restrictions; (b) each child gets at least 1₵;
(a) Find the generating function for the number of ways to select 10 candy bars from large supplies of six different kinds.(b) Find the generating function for the number of ways to select, with
(a) Explain why the generating function for the number of ways to have n cents in pennies and nickels is (1 + x + x2 + x3 +------)(1 + x5 + x10 + ......). (b) Find the generating function for the
Find the generating function for the number of integer solutions to the equation c1 + c2 + c3 + C4 = 20 where -3 < c1, -3 < c2, -5 < c3 < 5, and 0 < C4.
For S = {a, b, c}, consider the function f(x) = (1 + ax)( 1 + bx)( 1 + cx) = 1 + ax + bx + cx + abx2 + acx2 + bcx2 + abcx3. Here, in f(x) • The coefficient of x° is 1 -for the subset 0 of S. •
Find generating functions for the following sequences. [For example, in the case of the sequence 0, 1, 3, 9, 27, , the answer required is x/{ - 3x), not 3*;ci+1 or simply 0 + * + 3x2 + 9x3 +
In how many ways can two dozen identical robots be assigned to four assembly lines with (a) at least three robots assigned to each line? (b) at least three, but no more than nine, robots assigned to
In how many ways can 3000 identical envelopes be divided, in packages of 25, among four student groups so that each group gets at least 150, but not more than 1000, of the envelopes?
Two cases of soft drinks, 24 bottles of one type and 24 of another, are distributed among five surveyors who are conducting taste tests. In how many ways can the 48 bottles be distributed so that
If a fair die is rolled 12 times, what is the probability that the sum of the rolls is 30?
Carol is collecting money from her cousins to have a party for her aunt. If eight of the cousins promise to give $2, $3, $4, or $5 each, and two others each give $5 or $10, what is the probability
In how many ways can Traci select n marbles from a large supply of blue, red, and yellow marbles (all of the same size) if the selection must include an even number of blue ones?
How can Mary split up 12 hamburgers and 16 hot dogs among her sons Richard, Peter, Christopher, and James in such a way that James gets at least one hamburger and three hot dogs, and each of his
Verify that (1 - x - x2 - x3 - x4 - x5 - x6)-1 is the generating function for the number of ways the sum n, where n e N, can be obtained when a single die is rolled an arbitrary number of times.
Show that (1 - 4x)-1/2 generates the sequence (2nn), n ∈ N.
(a) If a computer generates a random composition of 8, what is the probability the composition is a palindrome?(b) Answer the question in part (a) after replacing 8 by n, a fixed positive integer.
Determine the sequence generated by each of the following generating functions.(a) f(x) = (2x - 3)3 (b) f(x) = x4/(l - x)(c) f(x) = x3/(l - x2) (d) f(x) = 1/(1 + 3x)(e) f(x) = 1/(3 -x)(f) f(x) = 1/(1
(a) How many palindromes of 11 start with 1? with 2? with 3? with 4?(b) How many palindromes of 12 start with 1? with 2? with 3? with 4?
Let n ∈ Z+, n odd. Can a palindrome of n have an even number of summands?
Let n ∈ Z+, n even. How many palindromes of n have an even number of summands? How many have an odd number of summands?
Determine the number of palindromes of n, where all summands are even, for (a) n = 10; (b) n = 12; and (c) n even.
Shay rolls a fair die until she gets a 6. If the random variable Y counts the number of times Shay rolls the die until she gets her first 6, determine (a) the probability distribution for F; (b)
Referring back to the preceding exercise, what is the probability Shay rolls her first 6 on an even-numbered roll?
Leroy has a biased coin where Pr(H) = 2/3 and Pr(T) = 1/3. Assuming that each toss, after the first, is independent of any previous outcome, if Leroy tosses the coin until he gets a tail, what is the
If Y is a geometric random variable with E{Y) = 7/3 determine (a) Pr(Y = 3); (b) Pr(Y > 3); (c) Pr(Y > 5); (d) Pr{Y > 5|F > 3); (e) Pr(Y > 6|Y > 4); and (f) σγ.
Consider part (a) of Example 9.17. (a) Determine the differences for the inequalities that result from the subset {3, 6, 8, 15} of S, and verify that those differences add to the correct sum. (b)
In each of the following, the function f(x) is the generating function for the sequence a0, a1, a2, ..., whereas the sequence b0, b1, b2, ....... is generated by the function g(x). Express g(x) in
In how many ways can we select seven nonconsecutive integers from {1, 2, 3, ..., 50}?
Use the following summation formulas to simplify the expression for Ck in Example 9.19:
(a) Find the first four terms c0, c\, c2, and c3 of the convolutions for each of the following pairs of sequences.i) an = 1, bn = 1, for all n ∈ Nii) an = 1, bn = 2n, for all n ∈ Niii) a0 = a1 =
Find a formula for the convolution of each of the following pairs of sequences. (a) an = 1, 0 < n < 4, an = 0, for all n > 5; bn = n, for all n > N (b) an = (-1)n, bn = (-1)n, for all n ∈ N
(a) Find the coefficient of x7 in (l + x + x2 + x3 +......)15. (b) Find the coefficient of x1 in (1 + x + x2 + + ...... )n for n ∈ Z+.
For n ∈ Z+, find in (1 + x + x2)(1 + x)n the coefficient of (a) x7; (b) x8; and (c) xr for 0 < r < n + 2, r ∈ Z.
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