Concept explainers
Counting Methods. Answer the following questions us-
ing the appropriate counting technique. which may be either
arrangements with repetition. permutations. Or combinations.
Be sure to explain why this counting technique applies to the
problem.
23. HOW many different nine-digit ZIP codes can be formed?
24. How many different six-character can formed
from the lowercase letters of the ?
25. HOW many different six-character passwords can formed
from the lowercase letters of the alphabet if repetition is not
allowed?
26. A city council with eight members must elect a
executive committee consisting of a mayor, secretary, and
treasurer. How many executive committees are possible?
27. How many ways can the eight performances at a piano recital
be ordered?
28. A city council with ten members must appoint a four-person
subcommittee. How many subcommittees are possible?
29. Suppose you have 15 CDs from which you 6 CDs to
put in the CD player in your car. If you are not particular
about the order, how many O-CD sets are possible?
30.HOW many 6-person can be formed from a & player
volleyball assuming every player can be assign to
any position?
31. How many different birth orders with respect to gender
possible in a family with five children? (For example.
and BGBGG are different orders.)
32. HOW many different 5-cards can be dealt from a 52-card
deck?
33. How many license plates can be made of the form XX—YYYY,
where X is a letter Of the and Y is a numeral 0—9?
34. How many different groups of balls can drawn from
a barrel containing balls numbered 1—36?
35. How many different telephone numbers of the form aaa-bbb-
cccc formed if the area code cannot contain 0 and
the prefix bbb cannot contain 9?
36. HOW many anagrams (rearrangements) Of the letters
ILOVEMATH can nuke?
37. How many different three-letter “words”- can formed from
the ACGT?
38. The debate club has 18 members, but only 4 can compete
at the next meet. How many 4-Frson teams are possible?
39. A recording engineer wants to make a CD With 12 songs. In
how many different ways can the CD nude?
40. A shelter is giving away 15 but you have
room for only 4 of them. How many different families
could you have?
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Check out a sample textbook solutionChapter 7 Solutions
Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
- 6. [10 marks] Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of T. (a) How many vertices does BL(T) have? (b) How many edges does BL(T) have? Prove that your answers are correct.arrow_forward4. [10 marks] Find both a matching of maximum size and a vertex cover of minimum size in the following bipartite graph. Prove that your answer is correct. ย ພarrow_forward5. [10 marks] Let G = (V,E) be a graph, and let X C V be a set of vertices. Prove that if |S||N(S)\X for every SCX, then G contains a matching M that matches every vertex of X (i.e., such that every x X is an end of an edge in M).arrow_forward
- Q/show that 2" +4 has a removable discontinuity at Z=2i Z(≥2-21)arrow_forwardRefer to page 100 for problems on graph theory and linear algebra. Instructions: • Analyze the adjacency matrix of a given graph to find its eigenvalues and eigenvectors. • Interpret the eigenvalues in the context of graph properties like connectivity or clustering. Discuss applications of spectral graph theory in network analysis. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 110 for problems on optimization. Instructions: Given a loss function, analyze its critical points to identify minima and maxima. • Discuss the role of gradient descent in finding the optimal solution. . Compare convex and non-convex functions and their implications for optimization. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 140 for problems on infinite sets. Instructions: • Compare the cardinalities of given sets and classify them as finite, countable, or uncountable. • Prove or disprove the equivalence of two sets using bijections. • Discuss the implications of Cantor's theorem on real-world computation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 120 for problems on numerical computation. Instructions: • Analyze the sources of error in a given numerical method (e.g., round-off, truncation). • Compute the error bounds for approximating the solution of an equation. • Discuss strategies to minimize error in iterative methods like Newton-Raphson. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 145 for problems on constrained optimization. Instructions: • Solve an optimization problem with constraints using the method of Lagrange multipliers. • • Interpret the significance of the Lagrange multipliers in the given context. Discuss the applications of this method in machine learning or operations research. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forward
- Only 100% sure experts solve it correct complete solutions okarrow_forwardGive an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.arrow_forward3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.arrow_forward
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