Homework 10 Solution Section 4.2, 4.3.

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1 MATH 00 Homewor Homewor 0 Solution Section.,.3. Please read your writing again before moving to the next roblem. Do not abbreviate your answer. Write everything in full sentences. Write your answer neatly. If I couldn t understand it, you ll get 0 oint. You may discuss with your classmates. But do not coy directly.. There is a boo club consisting of 6 male students and female students. The members want to select a committee consisting of four students. a Find the number of ways to mae the committee b How many ways to mae a four-erson committee consisting of two male students and two female students are ossible? c How many ways to mae a four-erson committee with at least two female students are ossible? d How many ways to mae a four-erson committee consisting of a resident, a vice-resident, a secretary, and a treasurer? In this roblem, the order of selection matters. 8P 8!! 900. In an ad for Pizza Hut, Jessica Simson exlains to the Muets that there are more than 6 million ossibilities for their forall Pizza. Unfortunately, I was unabled to find the video cli of this commercial, but here is another version in 00: clic here to watch

2 MATH 00 Homewor a Each izza can have u to 3 toings, out of 7 ossible choices, or can be one of four secialty izzas. Calculate the number of different izzas ossible. 7 The number of izzas with toings is. Because there are seciality izzas, the total number is b Out of the total ossible izza calculated in the first art of this exercise, a forall Pizza consists of four izzas in a box you may order some same ind izzas. Calculate the total number of forall Pizzas ossible. Comare with her comutation. To mae a box, we have to choose four izzas. The order of selection is not imortant, and reetition is allowed. Therefore the number of selection is , 695, 8, 670, which is way larger than her comutation. 3. a Prove the following equality by using algebra. n n n. n n!!n! n!!n! n!!n! nn!!n! n n!!n! n b Prove the same equality by using combinatorial idea. n Consider the situation counting the number of ways to select a -erson committee, and a chair in the selected committee from a grou of n eole. We will count the number of ways in two different ways. n First of all, the number of ways to select such a committee is. The number of ways to select a chair is. Thus the total number of ways n is. On the other hand, we may select a chair first, and then select the remaining committee members. Then the number of ways to choose a chair is

3 MATH 00 Homewor n n, and the number of ways to select remaining members is. n Therefore the total number is n. These two numbers must be same because they are counting the same number of ways. Therefore n. n n. Prove the following equality by using combinatorial idea. r 0 r s n r s Suose that there are r men and s women. We will comute the number of ways r s to select n eole, without regarding the order. This number is. n On the other hand, in this selected grou, the number of men can be any integer from 0 to r. In the case that the number of men is, there must be n r s women, so the number of ways to choose a grou is. By adding all n of the numbers for 0,,, r, we can obtain the total number of selections. Therefore, r r s r s. n n 0 5. Let be a rime number. a Show that is a multile of for 0 < <. Note that!!!. On the right hand side, on the numerator, is a divisor. If 0,, then every factor on the denominator is an integer less than. Because is a rime number, none of the factors of denominator divides. Therefore it is a multile of. b By using induction on n, rove that n n. Hint: For the inductive ste with induction hyothesis, use binomial theorem on. Then aly a. Let n. Then n n 0. Because 0, this case is true. Suose that n case is true. Then. By binomial theorem, 3 i0 n i i

4 MATH 00 Homewor 0. By a, s on the right hand side are all multiles of. By induction i hyothesis, is also a multile of. Therefore it is a multile of. 6. Let n N. Prove that the number of artitions of n equals the number of artitions of n with recisely n arts. I recommend to list all relevant artitions for n 3,, 5, 6 and try to mae exlicit bijections. Let P n be the set of artitions of n, and let P n,n be the set of artitions of n with n arts. Define a function f : P n,n P n as the following. For a, a,, a n P n,n, consider a new sequence b, b,, b which is obtained from a, a,, a n by discarding all 0 s. Define fa, a,, a n b, b,, b. We claim that f is a bijective function. The simlest way to chec it is constructing its inverse function. Let g : P n P n,n be a function defined by gb, b,, b b, b,, b,,,, the result is a sequence of length n. It is straightforward to chec that g is the inverse of f. Since f is bijective, P n P n,n. 7. There are n eole resent in a room. Prove that among them there are two eole who have the same number of acquaintances in the room. Here is a urely grah theoretic interretation of the same roblem. Let G be a simle grah with n vertices. Prove that there are two vertices with the same degree. Let X be the set of n eole in the room and let Y {0,,,, n }. Define a function f : X Y as fx the number of acquaintances. Suose first, that there is no erson with no acquaintance. Then we may use Y {,,, n } as the codomain instead of Y. Then X n and Y n, so X > Y. By igeonhole rincile, f is not an injective function. So there are a, b X so that fa fb. This means there are two eole with the same number of acquaintances. Now suose that there is a erson x with no acquaintance. If there is a erson y with n acquaintances, then x and y must now each other. It contradicts to the fact that x has no acquaintance. Therefore there is no erson with n acquaintances. So we may relace the codomain Y by Y {0,,,, n }.

5 MATH 00 Homewor Then again, X > Y. So by the same argument, we can conclude that there are two eole with the same number of acquaintances. 8. Show that among 3 eole, there are two born in the same month. Let X be the set of those 3 eole and let Y be the set of months. Let f : X Y be a function that fx the month that x was born. Then X 3 and Y. By igeonhole rincile, there are a, b X such that fa fb. So there are two eole who were born in the same month. Below is a set of exercise roblems for Chater, including the last section.. They are not homewor. This list may be udated. 9. By using combinatorial idea, show that n r r for any three natural numbers r n. n n r 0. Recall that from the binomial theorem. x n n r0 n x r r a By using algebra and calculus, show that n n r n n. r r0 Hint: Tae the derivative of. b By using combinatorial idea, rove the same equality.. Let a, b be two corime natural numbers. Show that the decimal reresentation of a/b has at most eriod b.. A target has the form of an equilateral triangle with side. Prove that if it is hit 5 times, then there will be two holes with distance. 3. Suose that there are distinct two-digit natural numbers n, n,, n. Show that there are two numbers whose difference is of the form aa.. Let a, a,, a n be n integers. Prove that there always exists a subset of these numbers with sum divisible by n. 5

6 MATH 00 Homewor 5. A controversy arose in 99 over the Teen Tal Barbie doll, each of which was rogrammed with four sayings randomly iced from a set of 70 sayings. The controversy ways over the saying, Math class is tough, which some felt gave a negative message toward girls doing well in math. In an interview with Science, a soeswoman for Mattel, the maers of Barbie, said that There s a less than % chance you re going to get a doll that says math class is tough. Is this figure correct? If not, give the correct figure. 6. During the 009 season, the Washington Nationals baseball team won 59 games and lost 03 games. Season ticet holder Stehen Kruin reorted in an interview that he watched the team lose all 9 games that he attended that season. The interviewer seculated that this must be a record for bad luc. a Based on the full 009 season record, calculate the robability that a erson would attend 9 Washington Nationals games and the Nationals would lose all 9 games. b However, Mr. Kruin only attended home games. The Nationals has 33 wins and 8 losses at home in 009. Calculate the robability that a erson would attend 9 Washington Nationals home games and the Nationals would lose all 9 games. 7. Suose that 3 edges of K n are chosen at random. Find the robability that these edges form a triangle. 8. Suose that 3 eole are in a classroom. a Find the robability that their birthdays are all different. b Find the robability that there are two eole with the same birthday. 6

and their probabilities p

and their probabilities p AP Statistics Ch. 6 Notes Random Variables A variable is any characteristic of an individual (remember that individuals are the objects described by a data set and may be eole, animals, or things). Variables