Explanations for Discrete Structures problems
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11) Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most three action figures.
12) Use generating functions to find the number of ways to select 10 balls from an urn containing red, white and blue balls if:
a. The selection has at least two balls of each color.
b. The selection has at most two balls of each color.
c. The selection has an even number of red balls.
13) Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y) E R, if and only if:
a) x + y = 0
b) xy = 0
14) Find:
a) R1 U R3
b) R1 - R2
c) R2 (symmetric difference) R4
15) Show that the sum, over the set of people at a party, of the number of people a person has shaken hands with, is even.? Assume that no one shakes his or her own hand.
16) What is the sum of the entries in a column of the adjacency matrix for an undirected graph?? For a directed graph?
17) Show that the PETERSON GRAPH, does not have a Hamilton circuit, but that the subgraph obtained by deleting a vertex v, and all edges incident with v, does have a Hamilton circuit.
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Solution Summary
The solution contains detailed explanations of whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive. Some problems in graph theory are discussed as well.
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11). We can consider this question as follows. Labeling children 1,2,3,4,5.
1) Just four children have 3 action figures, so there are 5 cases.
2) three children have 3 action figures, there are 10 cases, but the rest two may have two or one , or one or two, so there are 10*2=20 cases.
3) two children have 3 action figures, there are 10 cases, but the rest three children must have two for each, so there are 10 cases.
2) one child has 3 action figure, then the rest four children have total 9. This is impossible. You can know this by pigeonhole principle.
So the total number is 5+20+10=35.
12. a) It is equivalent to choose 4 balls from urn containing red , white and blue balls. We consider as follows. Just consider ...
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- BSc , Wuhan Univ. China
- MA, Shandong Univ.
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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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