Abby, Bridget和其他四个同班同学坐在如下所示的两行三列的座位上准备拍集体照。 $X\ X\ X$
$X\ X\ X$ 如果座位是随机分配的，那么Abby和Bridget坐在同一行或者同一列的相邻位置上的概率是多少？

A. $\frac{1}{3}$

B. $\frac{2}{5}$

C. $\frac{7}{15}$

D. $\frac{1}{2}$

E. $\frac{2}{3}$

1. 2018 AMC 8 官方解析

method 1:

There are $6!$ ways to arrange all students. Abby and Bridget are adjacent to each other in the same column in these three arrangements:

$A\ X\ X\quad X\ A\ X\quad X\ X\ A$
$B\ X\ X\quad X\ B\ X\quad X\ X\ B$

Observe that Abby and Bridget can exchange seats without violating this condition, and that there are $4!$ ways to arrange the other students. Therefore, there are $3 \times 2 \times 4!$ ways for Abby and Bridget to be adjacent in the same column.

Similarly, there are four arrangements in which Abby and Bridget are adjacent to each other in the same row. They can exchange seats without violating this condition, and there are $4!$ ways to arrange the other students. Therefore, there are $4 \times 2 \times 4!$ ways for Abby and Bridget to be adjacent in the same row.Then probability is then,
$\frac{(3 \times 2) \times 4! + (4 \times 2) \times 4!}{6!} = \frac{14 \times 4!}{6!} = \frac{7}{15}$.The correct answer is C.

method 2:

In this method, we practically disregard the other students in our calculation. Let Abby and Bridget "stick together." There are $\binom{6}{2} = 15$ ways to get them seated. Moreover, there are $2 \times 2 = 4$ ways for them to be seated in the same row and three ways in the same column.
The probability is then, $\frac{4+3}{15} = \frac{7}{15}$.The correct answer is C.