题目信息
In the figure above, X and Y represent locations in a district of a certain city where the streets form a rectangular grid. In traveling only north or east along the streets from X to Y, how many different paths are possible?
A:720
B:512
C:336
D:256
E:56
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已选答案：
正确答案：
E:56
Arithmetic Elementary combinatorics
Each possible path will consist of traveling a total of 3 grid segments north and 5 grid segments east. Thus, letting ‘N' represent traveling north by one grid segment and ‘E' represent traveling east by one grid segment, each path can be uniquely represented by an appropriate 8-character string of N's and E's. For example, as shown in the figure below, NEENEENE represents grid segments traveled in the order north, east, east, north, east, east, north, and east.
Therefore, the number of possible paths is equal to the number of appropriate 8-character strings of N's and E's, which is
= (7)(8) = 56, since each appropriate string is determined when a specification is made for the 3 positions in the string at which the N's are to be placed. Alternatively, the number of possible paths is equal to the number of permutations of 8 objects in which 3 are identical (the N's) and the remaining 5 are identical (the E's), and thus equal to 
.
Tip The alternative approach has a well-known generalization that can be used to calculate the number of permutations of n objects when various subsets of those objects consist of objects to be treated as identical. We give four examples in which such a calculation can be used. The number of 8-letter words that can be formed using the letters of PEPPERER is equal to
= 560.
Consider a 3-dimensional analog of the rectangular grid above, with dimensions 2 by 3 by 3. The number of paths from the front-left-down vertex to the back-right-up vertex such that each path consists of traveling only back, right, or up is equal to = 560.
The number of ways to distribute 8 different books to David, Liam, and Sophia so that David is given 2 of the books, Liam is given 3 of the books, and Sophia is given 3 of the books is equal to = 560.
The coefficient of a2b3c3 in the expansion of (a + b + c)8 after like terms are combined is equal to 
= 560.
The correct answer is E.
Each possible path will consist of traveling a total of 3 grid segments north and 5 grid segments east. Thus, letting ‘N' represent traveling north by one grid segment and ‘E' represent traveling east by one grid segment, each path can be uniquely represented by an appropriate 8-character string of N's and E's. For example, as shown in the figure below, NEENEENE represents grid segments traveled in the order north, east, east, north, east, east, north, and east.
Therefore, the number of possible paths is equal to the number of appropriate 8-character strings of N's and E's, which is
= (7)(8) = 56, since each appropriate string is determined when a specification is made for the 3 positions in the string at which the N's are to be placed. Alternatively, the number of possible paths is equal to the number of permutations of 8 objects in which 3 are identical (the N's) and the remaining 5 are identical (the E's), and thus equal to .Tip The alternative approach has a well-known generalization that can be used to calculate the number of permutations of n objects when various subsets of those objects consist of objects to be treated as identical. We give four examples in which such a calculation can be used. The number of 8-letter words that can be formed using the letters of PEPPERER is equal to
= 560.
Consider a 3-dimensional analog of the rectangular grid above, with dimensions 2 by 3 by 3. The number of paths from the front-left-down vertex to the back-right-up vertex such that each path consists of traveling only back, right, or up is equal to = 560.
The number of ways to distribute 8 different books to David, Liam, and Sophia so that David is given 2 of the books, Liam is given 3 of the books, and Sophia is given 3 of the books is equal to = 560.
The coefficient of a2b3c3 in the expansion of (a + b + c)8 after like terms are combined is equal to = 560.The correct answer is E.
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