Đang tải... (xem toàn văn)
Eulers formula is considered a special formula in the high school math curriculum. In this book, we will introduce you to the definition as well as the applications of this formula in problem solving. The Euler candy division problem is a wellknown counting problem with many applications in other counting problems. In this lesson, I present the basic original problem and some applied counting problems that would be very difficult to count in the normal way, but when you understand the counting of Eulers problem, the problem becomes simple. Hope they help you to get high score
o c l e W o t m u o gr p Member of group 4: • Trương Khánh Huyền • Vũ Thị Khánh Linh • Nguyễn Thị Lan • Trần Đình Vũ Introduction Euler's formula is considered a special formula in the high school math curriculum In this book, we will introduce you to the definition as well as the applications of this formula in problem solving The Euler candy division problem is a well-known counting problem with many applications in other counting problems In this lesson, I present the basic original problem and some applied counting problems that would be very difficult to count in the normal way, but when you understand the counting of Euler's problem, the problem becomes simple Hope they help you to get high score! Problem: repeating combination and Euler's candy division problem I Repeating combination II Around Euler's candy division problem I.Repeating combination Problem: A person goes to a store to buy school supplies as a gift including pens, books, and notebooks, in total he only buys items Given that there are identical pens, books, and 10 notebooks in a store, how many ways are there to choose pens, books, and notebooks for a gift? We see that the number of pén, books and notebooks is greater than the quantity to be purchased, so the only problem returned to the counter is how many sets of lists are there for a total of 5, where each set has or has none There are three objects: pens, books, and notebooks, which we denote by A=P, B, N A gift consists of pieces, so the gift can be X= P, P, P, B, N consisting of pens and book, notebook, or a set of Y=P, P, B, N, N We can see that the objects P, N is repeated Then we say the combination of X, and Y is r-combination Definition: Let the set A = a1, a2, …, ak A mapping from p: A↦N, then P is called a multiset of A Proposition : Given the set A = a1, a2, …, ak the number of mappings p : A ↦ N such that p(a1) + p(a2) +⋯+ p(ak) is Prove Each mapping, we give corresponds to a binary sequence of length n+k−1, where p(a1) the first digit is 0, followed by 1, then p(a2) digit … and finally p(a) digit For example, the set of PPBNN corresponds to the sequence 0010100 This is a - correspondence, so the number of mappings pp is equal to the number of binary sequences, so we only need to count the number of binary sequences We see that the sequence has n+k−1 digits in which there are k−1 digits, so the number of binary sequences is just the number of ways to choose positions for k−1 digits, so the number of binary sequences is So the mapping number p is Going back to the above problem, we see that the number of gifts with is a convolutional combination of of books, writing, and books, so the number of possible gifts is = Note: In the above problem, make sure the number of each product type is not less than pieces II Around Euler's candy division problem Starting from a problem “There are n identical candies divided by m babies How many ways are there to divide the candy in all?” The math seems to be very simple but it is a difficult problem for many students There is a combinatorial problem in the National Examination for Good Students, the solution of which can be prevented by applying the results of Euler's candy division From the actual problem, deduce the results of Euler's candy division problem We all know that, in a binary sequence, the elements take the value or The number of valid binary sequences has length n and in each sequence, there are exactly k (0 ≤ k ≤ n) elements that take the value equal to The opening problem Given a grid of squares The nodes are numbered from to m from left to right and from to n from bottom to top How many different paths are there from the node (0,0) to node (m;n) if only the edges of the squares are allowed to go from left to right or from bottom to top? Solution : Such a path is considered to consist of (m+n) segments (each segment is a square edge) At each segment, only one of two values can be selected to go up (we encode 1) or to the right (we encode 0) The number of segments going up is exactly equal to n and the number of segments to the right is exactly m The problem leads to finding how many binary sequences of length ( m + n ) in which there are exactly n elements with the value The desired result is Euler's candy division problem There are n identical candies divided by m babies How many ways are there to divide the candy in all? Or the problem itself : Find the number of non-negative integer solutions of the equation x1 + x2 +…+ xm = n (m,nN) According to the opening problem, the number of tasks to find is ... formula in problem solving The Euler candy division problem is a well-known counting problem with many applications in other counting problems In this lesson, I present the basic original problem. .. Problem: repeating combination and Euler's candy division problem I Repeating combination II Around Euler's candy division problem I.Repeating combination Problem: A person goes to a store to buy... than pieces II Around Euler's candy division problem Starting from a problem “There are n identical candies divided by m babies How many ways are there to divide the candy in all?” The math seems