(a) Find all triples (a, b, c) of positive integers such that a! + b! =...
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(a) Find all triples (a, b, c) of positive integers such that a! + b! = c!, and justify your answer. (b) Given two points (a, b, c), (r, y, z) e R®, we say that the midpoint of these two points is the point with coordinates (,,). Take any set S of nine points from R* with integer coordinates (i.e. every coordinate in each of these points is an integer.) Prove that there must be at least one pair of points in S whose midpoint also has integer coordinates. (c) Three counting problems are described below. For each problem, do the following: find the answer, and then justify the method you used to count these objects. Give your answer as an integer; i.e. write 24 instead of 4! or (), i. You have ten indistinguishable cookies that you've baked for four of your friends Jianbei, Sione, Sina and Julia. You want to give away all ten of your cookies to your friends, and for each friend to get at least one cookie. You also know that Sione wants an even number of cookies, so he can share them with a friend. In how many ways can you give away your cookies? ii. You're a Pokémon master! You have collected exactly one of all 151 Pokémon in the base game. You want to assemble a team to take on the Elite Four: doing this involves choosing a team of 6 Pokémon (in which the order does not matter) and then designating one of those six to be the "lead" on your team (i.e. the first one to go out.) In how many ways can you do this? iii. Baby names! There are tons of names out there! In an effort to cut down the number of names to think about, suppose that you had the following restrictions': • You want a name that starts with the first letter of your last name (alliteration!) • You want a name between 4-5 letters (quick to write.) • You think that the letter "z" is cool, and want exactly one "z" in the name. How many names exist that meet all of these restrictions? (a) Find all triples (a, b, c) of positive integers such that a! + b! = c!, and justify your answer. (b) Given two points (a, b, c), (r, y, z) e R®, we say that the midpoint of these two points is the point with coordinates (,,). Take any set S of nine points from R* with integer coordinates (i.e. every coordinate in each of these points is an integer.) Prove that there must be at least one pair of points in S whose midpoint also has integer coordinates. (c) Three counting problems are described below. For each problem, do the following: find the answer, and then justify the method you used to count these objects. Give your answer as an integer; i.e. write 24 instead of 4! or (), i. You have ten indistinguishable cookies that you've baked for four of your friends Jianbei, Sione, Sina and Julia. You want to give away all ten of your cookies to your friends, and for each friend to get at least one cookie. You also know that Sione wants an even number of cookies, so he can share them with a friend. In how many ways can you give away your cookies? ii. You're a Pokémon master! You have collected exactly one of all 151 Pokémon in the base game. You want to assemble a team to take on the Elite Four: doing this involves choosing a team of 6 Pokémon (in which the order does not matter) and then designating one of those six to be the "lead" on your team (i.e. the first one to go out.) In how many ways can you do this? iii. Baby names! There are tons of names out there! In an effort to cut down the number of names to think about, suppose that you had the following restrictions': • You want a name that starts with the first letter of your last name (alliteration!) • You want a name between 4-5 letters (quick to write.) • You think that the letter "z" is cool, and want exactly one "z" in the name. How many names exist that meet all of these restrictions?
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Related Book For
Probability and Statistics
ISBN: 978-0321500465
4th edition
Authors: Morris H. DeGroot, Mark J. Schervish
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