We want to build a chatbot, so we put together a list of 10 nouns, 10 verbs, and 10 objects, so we can construct sentences of the form NOUN VERB OBJECT (e.g., "Everyoneloves combinatorics"). But we notice that 3 of the words on the noun list also appear on the verb list. None of the words on the object list appear on either of the other twolists.How many sentences of the form NOUN VERB OBJECT can we make if we require that the NOUN and VERB be different words?Type your answer

We can use combinations to answer this question.

Answered: We want to build a chatbot, so we put…

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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An instructor gives an exam with twenty questions. Students are allowed to choose any fifteen questions to answer. (a) How many different choices of fifteen questions are there? (b) Suppose nine questions require coding and eleven do not. (i) How many groups of fifteen questions contain seven that require coding and eight that do not? (ii) How many groups of fifteen questions contain at most eight that require coding? (c) Suppose the exam instruction specifies that at most one of questions 1 and 2 MAY BE included among the fifteen. How many different choices of fifteen questions are there? (d) Suppose the exam instruction specifies that either both questions 1 and 2 are to be included among the ten or neither is to be included. How many different choices of ten questions are there?
A number in Z+is written on the back of each of three cards. The numbers are not necessarily different but the sum of the three numbers is 14. Each card is flipped backwards so that the number cannot be seen by anyone. Carter, Mars, and Mann each select one of the cards, being careful not to let the other two see the number that is on the card. Carter says, after looking at his card, “I know that Mars and Mann have different numbers.” Mars then says, “I already knew that thet hree cards had different numbers.” At this point, Mann confidently exclaims, “I now know all thet hree numbers!” What were the numbers and who had which number?
A string that contains only Os, 1s, and 2s is called a ternary string. How many ternary strings of length 9 start with a "2" and end with a "1" but don't have a "0" in the third position?
We have 3 Physics books, 4 Math books, and 5 Chemistrybooks. The 12 books are distinct. We want to arrange them all on abookshelf. Observe that without any restrictions there are 12! ways to arrangethem on a bookshelf. Explain each answer. In how many ways can we do this(a) if the arrangement must start with a Physics book and must end with aChemistry book?(b) if the arrangement cannot end with a chemistry book?(c) if the arrangement must begin with all the Physics books?(d) if the two specific books X and Y (of the 12 books) cannot be next toeach other?(e) if all the books belonging to a subject must stay together (i.e. stay nextto each other)? (hint: The subjects themselves have to be in someorder.)
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 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?
Suppose we want to choose 5 objects, without replacement, from 9 distinct objects. (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?
We want to divide 12 children into any number of playgroups. However, in order to avoid some predictable disagreements, we want Adam and Brenda to be in different playgroups. How many ways are there to do this? (Playgroups have at least one child)
ABCbooks, an internet book retailer, is interested in better understanding the purchase decisions of its customers. For a set of 2,000 customer transactions, it has categorized the individual book purchases comprising those transactions into one or more of the following categories: Novels, Cooking Books, Do-It-Yourself, Business, Art Books, Biography, and Harry Potter series. Table below shows support for Antecedent and Consequent of 5 rules. From these 5 rules, what is the most useful rule? Justify your answer with logical reasons and appropriate calculations.
Use this for 11-12: A student club consists of 10 girls and 10 guys. A committee of 8 is to be selected. In how many ways can this be done if (page 168- 172) Show your work ! 11.) the committee must have at least 7 girls on it12.) three officers (president, vice president, secretary/treasurer) are selected from ALL the members.

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