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For convenience of notation we introduce the following convention. Any finite sequence of English letters will be called "a word" whether or not it can be found in a dictionary. In the following five problems you must calculate the number of different words that can be obtained by rearranging the letters of a particular word.

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Since all the letters in this word are different, the answer is 6! This word contains two letters T, and all the other letters are different. Let us temporarily think of these letters T as two different letters Tl and T2. Under this assumption we have 5! However, any two words which can be obtained from each other just by transposing the letters Tl and T2 are, in fact, identical.

Thus, our words split into pairs of identical words. However, any words which can be obtained from each other just by transposing the letters A, are identical. Since the letters Ai can be rearranged in their places in 3! Therefore the answer is 8! We have three letters S and two letters E in this word.

Temporarily thinking of all of them as different letters, we have 9! Then, recalling that the letters S are identical, we come to the final answer: 9! This set of problems about words demonstrates one very interesting and important idea-the idea of multiple counting. That is, instead of counting the number of objects we are interested in, it may be easier to count some other objects whose number is some known multiple of the number of objects.

Here are four more problems using this method. There are 20 towns in a certain country, and every pair of them is connected by an air route. How many air routes are there? Every route connects two towns. We can choose any of the 20 towns in the country say, town A as the beginning of a route, and we have 19 remaining towns to choose the end of a route say, town B from. However, this calculation counted every route AB twice: when A was chosen as the beginning of the route, and when B was chosen as the beginning. A similar problem is discussed in the chapter "Graphs-I" where we count the number of edges of a graph.

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How many diagonals are there in a convex n-gon? Any of the n vertices can be chosen as the first endpoint of a diagonal, and we have n - 3 vertices to choose from for the second end any vertex, except the chosen one and its two neighbors. Counting the diagonals this way, we have counted every diagonal exactly twice. See Figure A "necklace" is a circular string with several beads on it.

It is allowed to rotate a necklace but not to turn it over. How many different necklaces can be made using 13 different beads? Let us first assume that it is prohibited to rotate the necklace. Then it is clear that we have 13! However, any arrangement of beads must be considered identical to those 12 that can be obtained from it by rotation.

Answer: 13! Assume now that it is allowed to turn a necklace over. How many necklaces can be made using 13 different beads? Answer: 12! How many six-digit numbers have at least one even digit? Instead of counting the numbers with at least one even digit, let us find the number of six-digit numbers that do not possess this property. The main idea in this solution was to use the method of complements; that is, counting or, considering the "unrequested" objects instead of those "reguested" Here is another problem which can be solved using this method.

There are six letters in the Hermetian language. A word is any sequence of six letters, some pair of which are the same. How many words are there in the Hermetian language? We also recommend reviewing the material already covered in previous sessions.

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For this reason we present here a list of problems for independent solution and for homework. In addition, you can take problems from [49] or create them yourself. Problems for independent solution Problem There are five types of envelopes and four types of stamps in a post office. How many ways are there to buy an envelope and a stamp? Seven nouns, five verbs, and two adjectives are written on a blackboard. We can form a sentence by choosing one word of each type, and we do not care about how much sense the sentence makes.

How many ways are there to do this? Each of two novice collectors has 20 stamps and 10 postcards. We call an exchange fair if they exchange a stamp for a stamp or a postcard for a postcard. How many ways are there to carry out one fair exchange between these two collectors? Problem , How many six-digit numbers have all their digits of equal parity Call odd or all even? In how many. How many ways are there to choose four cards of different suits and different values from a deck of 52 cards?

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There are five books on a shelf. How many ways are there to arrange some or all of them in a stack? The stack may consist of a single book. How many ways are there to put eight rooks on a chessboard so that they do Rot attack each other? There are N boy. How many ways are there to arrange them in pairs for a dance?

The rules of a chess tournament say that each contestant must play every other contestant exactly once. How many games will be played if there are 18 participants? How many ways are there to place a two bishops; b two knights; c two queens on a chessboard so that they do not attack each other? Mother has two apples, three pears, and four oranges.

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  6. Every morning, for nine days, she gives one fruit to her son for breakfast. There are three rooms in a dormitory: one single, one double, and one for four students.


    How many ways are there to house seVen students in these rooms? How many ways are there to place a set of chess pieces on the first row of a chessboard? The set consists -of a king, a queen, two identical rooks, two identical knights, and two identical bishops. How many "words" can be written using exactly five letters A and no more than three letters B and 10 other letters?

    How many ten-digit numbers have at least two equal digits? Problem 45!

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    We toss a die three times. Among all possible outcomes, how many have at least one occurrence of six? How many ways are there to split 14 people into seven pairs?

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    Problem 4B! How many nine-digit numbers have an even sum of their digits? This theme is not so recreational as some others, yet it contains large amounts of important theoretical material. Try to introduce elements of play in your sessions.

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    Even very routine problems like the factoring of integers can be turned into a contest by asking "Who can factor this huge number first? Since divisibility also enters into the school curriculum, you can use the knowledge acquired by students there. Prime and composite numbers Among natural numbers we can distinguish prime and composite numbers. A number is composite if it is equal to the product of two smaller natural numbers. Otherwise, and if the number is not equal to 1, it is called prime. The number 1 is neither prime nor composite.

    Prime numbers are like "bricks", which you can use to construct all natural numbers. How can this be done? Let us consider the number It is certainly composite. It can be represented, for instance, as 42 But each of the numbers 42 and 10 is composite, too.