AMC8 2014

AMC8 2014 · Q11

AMC8 2014 · Q11. It mainly tests Combinations, Inclusion–exclusion (basic).

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?

杰克想骑车从他家到吉尔家，吉尔家位于杰克家以东三条街区、以北两条街区的地方。每骑完一条街区，杰克可以选择继续向东或向北，但他需要避开一个位于他家以东一条街区、以北一条街区的危险路口。杰克骑行总共五条街区，他有多少种方式可以到达吉尔家？

(A) 4 4

(B) 5 5

(D) 8 8

(E) 10 10

Answer

Correct choice: (A)

正确答案：（A）

Solution

We can apply complementary counting and count the paths that DO go through the blocked intersection, which is $\dbinom{2}{1}\dbinom{3}{1}=6$. There are a total of $\dbinom{5}{2}=10$ paths, so there are $10-6=4$ paths possible. $\boxed{(\text{A})4}$ is the correct answer.

我们可以使用补集计数法，先计算经过被封路口的路径数，即 $\dbinom{2}{1}\dbinom{3}{1}=6$ 条。所有路径总数为 $\dbinom{5}{2}=10$ 条，所以可行路径为 $10-6=4$ 条。正确答案是 $\boxed{(\text{A})4}$。

Topics