(a) The number of subsets of an 100-set which are larger (strictly larger cardinal) thantheir complements.(b) The number of ways to paint 49 chairs using the colors red, blue, and green such thatboth the number of red chairs and the number of blue chairs is odd.(c) The number of ways to divide 7 children into any number of playgroups such thatAdam and Brenda be in different playgroups.(d) The number of positive integers not larger than 30000 that are divisible by at leastone of 2, 3, and 5.

I'll provide solutions to each part, along with explanations and justifications: a) The number of…

Answered: (a) The number of subsets of an 100-set…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose we want to choose 6 objects, without replacement, from 8 distinct objects. (If necessary, consult a list of formulas.) (a) If the order of the choices is not taken into consideration, how many ways can this be done? (b) If the order of the choices is taken into consideration, how many ways can this be done?
Suppose we want to choose 7 objects, without replacement, from 11 distinct objects.(If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices is taken into consideration? (b) How many ways can this be done, if the order of the choices is not taken into consideration?
16. Hotel Cardinality (formerly California) (H). It is the stranded traveler's fantasy. The Hotel Cardinality is a full-service luxury hotel with bar and restaurant. It has as many rooms as there are natural numbers. The room numbers are 1, 2, 3, 4, 5, . ... You can see why stranded travelers love the Hotel Cardinality. There appears to be no need for the sad sign: No Vacancy. Suppose, however, that every room is occupied. Now it appears that the night manager must flash the No Vacancy sign. What if a weary traveler were to arrive late at night looking for a place to stay? Could the night manager figure out a way to provide the traveler with a private room (no sharing) without evicting another guest? The answer is yes. Describe how this accommodation can be made; of course, some guests will have to move to other rooms.
Suppose that you had infinitely many people, each one wearing a uniquely numbered button: 1, 2, 3, 4, 5, .... You also have lots of pennies. You give each person a penny; then ask everyone to flip his or her penny at the same time. Then you ask them to shout out in order what they flipped (H for heads and T for tails). So you might hear: HHTHHTTТТHTTHTHТHTHHНННТН... or THTTTTHTHHHTTHTHTHTTTTTHHT... and so forth. Consider the set of all possible outcomes of their flipping (all possible sequences of H's and T's). Does the set of possible outcomes have the same cardinality as the natural numbers? Justify your answer.
HELP WITH PAR D AND PART F AND PART G (THOSE THREE PARTS)
Draw a "bag diagram" to illustrate the following set and find it cardinal number {{1,2}, {1,2,3}, {4}}
A set has a possibility of 32 subsets. What is the cardinality of the set? A set has a cardinality of 7. How many proper subsets does the set have? Given a set has 7 elements, how many subsets have 3 elements? Given a set has 15 elements, how many subsets have 7 elements? A set has a cardinality of 12. The number of subsets of 5 elements is equal to the number of subsets of ______ elements. Given the set {dog, cat, horse, hamster, turtle, fish, ferret, snake}, how many subsets contain 4 pets? A set has 45 subsets with 2 elements, what is the cardinality of the set?
5.) How many different 4-letter radio station call letters are possible if: (a) The first letter must be K or W and no letter may be repeated? (b) The first letter must be K or W, and repetition of letters is allowed? (c) The first letter must be K or W, no letter may be repeated, and the last letter is R? !
Suppose we want to choose 6 objects, without replacement, from 9 distinct objects. (If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices matters? ☐ (b) How many ways can this be done, if the order of the choices does not matter? ☐
Suppose we want to choose 7 objects, without replacement, from 11 distinct objects.(If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices is taken into consideration? (b) How many ways can this be done, if the order of the choices is not taken into consideration?
Suppose we want to choose 6 objects, without replacement, from 10 distinct objects. (If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices does not matter? ☐ (b) How many ways can this be done, if the order of the choices matters?
Suppose we want to choose 4 objects, without replacement, from 17 distinct objects. (a) If the order of the choices is not relevant, how many ways can this be done? (b) If the order of the choices is relevant, how many ways can this be done?

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