题目信息
5.Pat will walk from intersection X to intersection Y along a route that is confined to the square grid of four streets and three avenues shown in the map above. How many routes from X to Y can Pat take that have the minimum possible length?
A:Six
B:Eight
C:Ten
D:Fourteen
E:Sixteen
参考答案及共享解析
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本题耗时：
已选答案：
正确答案：
C:Ten
Arithmetic Elementary combinatorics
Each minimum-length route will consist of traveling a total of 3 grid segments up and 2 grid segments right. Thus, letting ‘U' represent traveling up by one grid segment and ‘R' represent traveling right by one grid segment, each minimum-length route can be uniquely represented by an appropriate 5-character string of U's and R's. For example, URUUR represents grid segments traveled in the order up, right, up, up, and right. Therefore, the number of possible minimum-length routes is equal to the number of appropriate 5-character strings of U's and R's, which is
= 10, since each appropriate string is determined when a specification is made for the 3 positions in the string at which the U's are to be placed.
Tip The ideas involved in counting the number of 5-character strings of U's and R's can be extended to counting the number of n-character strings when various characters being used are the same. For example, MISSISSIPPI has one M, four I's, four S's, and two P's, and thus the number of 11-letter words that can be formed using the letters of MISSISSIPPI is equal to
. (Each of the two expressions we have given for 34,650 is intended to suggest a method for counting the number of 11-letter words.)
The correct answer is C.
Each minimum-length route will consist of traveling a total of 3 grid segments up and 2 grid segments right. Thus, letting ‘U' represent traveling up by one grid segment and ‘R' represent traveling right by one grid segment, each minimum-length route can be uniquely represented by an appropriate 5-character string of U's and R's. For example, URUUR represents grid segments traveled in the order up, right, up, up, and right. Therefore, the number of possible minimum-length routes is equal to the number of appropriate 5-character strings of U's and R's, which is
= 10, since each appropriate string is determined when a specification is made for the 3 positions in the string at which the U's are to be placed.Tip The ideas involved in counting the number of 5-character strings of U's and R's can be extended to counting the number of n-character strings when various characters being used are the same. For example, MISSISSIPPI has one M, four I's, four S's, and two P's, and thus the number of 11-letter words that can be formed using the letters of MISSISSIPPI is equal to
. (Each of the two expressions we have given for 34,650 is intended to suggest a method for counting the number of 11-letter words.)The correct answer is C.
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