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Problem 888

AIME late

Combinatorics Difficulty 5.6 Find the answer

【Question 15】
There are 6 different roads from location A to location B. On the first day, Xiaofang travels from A to B and then returns from B to A; on the second day, he travels from A to B again and then returns to A. Each time Xiaofang returns from B to A, he does not take the same road he used to go from A to B. Therefore, the total number of different ways Xiaofang can travel back and forth between A and B over these two days is $\qquad$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

【Analysis and Solution】
(1) If the routes from location A to location B on the first and second days are the same, there are 6 choices for the route from A to B on these two days;
There are 5 choices for the return trip from B to A on the first day;
There are 5 choices for the return trip from B to A on the first day;
There are $6 \times 5 \times 5=150$ different ways;
(2) If the routes from A to B on the first and second days are different,

There are 6 choices for the route from A to B on the first day, and 5 choices for the route from A to B on the second day;
There are 5 choices for the return trip from B to A on the first day;
There are 4 choices for the return trip from B to A on the second day;
There are $6 \times 5 \times 5 \times 4=600$ different ways;
In summary, there are $150+600=750$ different ways for Xiao Fang to travel back and forth between A and B over these two days.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.