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About 11892 results. 1294 free access solutions
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| m97233 | The Tuesday night dance club is made up of six married couples and two of these twelve members must be chosen to find a dance hall for an upcoming fund raiser.
(a) If the two members are selected at random, what is the probability they are both women?
(b) If Joan and Douglas are one of the couples in the club, what is the probability at least one of them is among the two who are chosen? |
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| m97257 | Theorem 3.8 assumes that all calculations for det(A) are done by exact arithmetic. As noted previously, this is usually not the case in software. Hence, computationally, the determinant may not be a valid test for nonsingularity. Perform the following experiment: Let
Show that det(A) is 0, either by hand or by using your software. Next, show by hand computation that det(B) = -3ϵ, where
Hence, theoretically, for any ϵ ( 0, matrix B is nonsingular. Let your software compute det(B) for ϵ = (10-k, k = 5, 6, ( ( ( , 20. Do the computational results match the theoretical result? If not, formulate a conjecture to explain why not. |
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| m97260 | There are 132 ways in which one can parenthesize the product abcdefg.
(a) Determine, as in part (c) of Example 1.43, the list of five l s and five 0 s that corresponds to each of the following.
(i) (((ab)c)(d(ef)))
(ii) (a(b(c(d(ef)))))
(iii) ((((ab)(cd))e)f)
(b) Find, as in Example 1.43, the way to parenthesize abcdef that corresponds to each given list of five l s and five 0 s.
(i) 1110010100
(ii) 1100110010
(iii) 1011100100 |
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| m97261 | There are 15 people at a party. Is it possible for each of these people to shake hands with (exactly) three others? |
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| m97263 | There are b4 (= 14) ways to arrange 1, 2, 3, . . . , 8 in two rows of four so that (1) the integers increase in value as each row is read, from left to right, and (2) in any column the smaller integer is on top. Find, as in part (d) of Example 1.43,
(a) The arrangements that correspond to each of the following.
(i) 10110010
(ii) 11001010
(iii) 11101000
(b) The lists of four l s and four 0 s that correspond to each of these arrangements of 1, 2, 3, ... , 8.
(i) 1 3 4 5
2 6 7 8
(ii) 1 2 3 7
4 5 6 8
(iii) 1 2 4 5
3 6 7 8 |
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| m97271 | This exercise outlines a proof of the Birkhoff-von Neumann Theorem.
(a) For n ∈ Z+, an n × n matrix is called a permutation matrix if there is exactly one 1 in each row and column, and all other entries are 0. How many 5 × 5 permutation matrices are there? How many n × n?
b) An n × n matrix B is called doubly stochastic if btJ > 0 for all 1 < i < n, 1 < j < n, and the sum of the entries in each row or column is 1. If
verify that B is doubly stochastic.
(c) Find four positive real numbers c1, c2, c3, and c4, and four permutation matrices P1, P2, P3, and P4, such that c1 + c2 + c3 + c4 = 1 and B = c1P1 + c2P2 + C3P3 + c4P4.
(d) Part (c) is a special case of the Birkhoff-von Neumann Theorem: If B is an n × n doubly stochastic matrix, then there exist positive real numbers c1, c2, . . . , ck and permutation matrices P1, P2, . . . , Pk such that ∑ki=1 ci = 1 and ∑ki=1. To prove this result, proceed as follows: Construct a bipartite graph G = (V, E) with V partitioned as X ∪ Y, where X = {x1, x2, . . . , xn] and Y = {y1, y2,......, yn}- The vertex xn for all 1 < i < n, corresponds with the i th row of B; the vertex yj, for all 1 < j < n, corresponds with the j th column of B. The edges of G are of the form {xi, yj} if and only if bij > 0. We claim that there is a complete matching of X into Y.
If not, there is a subset A of X with |A| > |R(A)|. That is, there is a set of r rows of B having positive entries in s columns and r > s. What is the sum of these r rows of P? Ye |
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| m97284 | Three small towns, designated by A, B, and C, are interconnected by a system of two-way roads, as shown in Fig. 1.4.
(a) In how many ways can Linda travel from town A to town C?
(b) How many different round trips can Linda travel from town A to town C and back to town A?
(c) How many of the round trips in part (b) are such that the return trip (from town C to town A) is at least partially different from the route Linda takes from town A to town C? (For example, if Linda travels from town A to town C along roads R1 and R6, then on her return she might take roads R6 and R3, or roads R7 and R2, or road R9, among other possibilities, but she does not travel on roads R6 and R1.) |
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| m97288 | Three types of foam are tested to see if they meet specifications. Table 3.5 summarizes the results for the 125 samples tested.
Table 3.5
Let A, B denote the events
A: The sample has foam type 1.
B: The sample meets specifications.
Determine Pr(A), Pr(B), Pr{A ∩ R), Pr(A ∪ B), Pr(), Pr(), Pr( ∪ ), Pr( ∩ ), Pr(A ∆ B). |
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| m97292 | To raise money for a new municipal pool, the chamber of commerce in a certain city sponsors a race. Each participant pays a $5 entrance fee and has a chance to win one of the different-sized trophies that are to be awarded to the first eight runners who finish.
(a) If 30 people enter the race, in how many ways will it be possible to award the trophies?
(b) If Roberta and Candice are two participants in the race, in how many ways can the trophies be awarded with these two runners among the top three? |
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| m97300 | Travis tosses a fair coin twice. Then he tosses a biased coin, one where the probability of a head is 3/4, four times. What is the probability Travis s six tosses result in five heads and one tail? |
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| m97305 | Twelve clay targets (identical in shape) are arranged in four hanging columns, as shown in Fig. 1.5. There are four red targets in the first column, three white ones in the second column, two green targets in the third column, and three blue ones in the fourth column. To join her college drill team, Deborah must break all 12 of these targets (using her pistol and only 12 bullets) and in so doing must always break the existing target at the bottom of a column. Under these conditions, in how many different orders can Deborah shoot down (and break) the 12 targets? |
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| m97306 | Twelve patrons, six each with a $5 bill and the other six each with a $10 bill, are the first to arrive at a movie theater, where the price of admission is five dollars. In how many ways can these 12 individuals (all loners) line up so that the number with a $5 bill is never exceeded by the number with a $10 bill (and, as a result, the ticket seller is always able to make any necessary change from the bills taken in from the first 11 of these 12 patrons)? |
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| m97307 | Twelve points are placed on the circumference of a circle and all the chords connecting these points are drawn. What is the largest number of points of intersection for these chords? |
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| m97311 | Twenty-five slips of paper, numbered 1, 2, 3, ... , 25, are placed in a box. If Amy draws six of these slips, without replacement, what is the probability that
(a) The second smallest number drawn is 5?
(b) The fourth largest number drawn is 15?
(c) The second smallest number drawn is 5 and the fourth largest number drawn is 15? |
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| m97317 | 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 each surveyor gets
(a) at least two bottles of each type?
(b) at least two bottles of one particular type and at least three of the other? |
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| m97329 | Two integers are selected, at random and without replacement, from {1, 2, 3, . . . , 99, 100}. What is the probability their sum is even? |
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| m97330 | Two n-digit integers (leading zeros allowed) are considered equivalent if one is a rearrangement of the other. (For example, 12033, 20331, and 01332 are considered equivalent five-digit integers.)
(a) How many five-digit integers are not equivalent?
(b) If the digits 1,3, and 7 can appear at most once, how many nonequivalent five-digit integers are there? |
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| m97350 | Use a complete ternary decision tree to repeat Example 12.15 for a set of 12 coins, exactly one of which is heavier (and counterfeit). |
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| m97355 | Use a Karnaugh map to find a minimal-sum-of products representation for
(a) f(w, x, y, z) = ∑ m(0, 2, 3, 6, 7, 14, 15)
(b) g(v, w, x, y, z) = Π M(1, 2, 4, 6, 9, 10, 11, 14, 17, 18, 19,20, 22, 25,26, 27, 30) |
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| m97359 | Use a procedure similar to the one discussed after Example 3 to develop sine and cosine expressions for the difference of two angles; θ1 - θ2. |
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