On the math parking problem

Problem 1: There are 10 parking spaces in a line adjacent to one another, and 3 identical cars come to the parking lot. They do not wish to park next to any other car. How many ways can they do this?

Problem 2: There are 8 parking spaces in a line adjacent to one another, and 3 identical cars come to the parking lot. How many ways can they do this?

Note: In Problem 2, there is no restriction "They do not wish to park next to any other car".

It is easy to solve Problem 2, and the answer is C(8,3) = 56.

We will argue that Problems 1 and 2 have the same answer by establishing a one-to-one correspondence between the ways of parking in Problem 1 and the ways of parking in Problem 2. For each way of parking in Problem 1, taking away one parking space from the left of the middle car and another one from the right yields one way of parking in Problem 2. Conversely, for each way of parking in Problem 2, adding one parking space to the left of the middle car and another one to the right yields one way of parking in Problem 1. Hence, the answer to Problem 1 is also 56.

所有跟帖： 

• OK. This is nice. -nj_guy- ♂ (211 bytes) () 10/16/2011 postreply 13:30:31

• 你问题在于， -1妈- ♀ (659 bytes) () 10/16/2011 postreply 14:20:48

• 那以后专砸你吧!咱两熟:) -nj_guy- ♂ (0 bytes) () 10/16/2011 postreply 14:24:49

• 没问题，砸别人的金砖俺也接。。。 -1妈- ♀ (113 bytes) () 10/16/2011 postreply 14:27:23

• 那还是算了,金砖太贵,咱砸不起. -nj_guy- ♂ (0 bytes) () 10/16/2011 postreply 14:34:44

• C(12,2)-10=56 is another way. -1妈- ♀ (0 bytes) () 10/16/2011 postreply 14:13:46

• 笨办法：c(10,3)-c(9,2)-c(8,1)=56 -ttmum- ♀ (189 bytes) () 10/16/2011 postreply 19:10:14

• What if 10 is replaced by 25 and 3 by 7? -BeLe- ♀ (0 bytes) () 10/16/2011 postreply 19:21:26

• 计算太麻烦了：） -ttmum- ♀ (155 bytes) () 10/16/2011 postreply 19:44:08

• 回复：计算太麻烦了：） -BeLe- ♀ (55 bytes) () 10/16/2011 postreply 20:41:18

• ic....容我回头再想想去，thx:-) -ttmum- ♀ (0 bytes) () 10/16/2011 postreply 20:49:29

• 看看这里。 -BeLe- ♀ (126 bytes) () 10/16/2011 postreply 21:34:32

• c(10,3)-c(9,2)-c(8,1)=76 -1妈- ♀ (0 bytes) () 10/16/2011 postreply 19:22:25

• 我记错公式了... -ttmum- ♀ (29 bytes) () 10/17/2011 postreply 16:49:40

• C(10,3) - [P(9,2)-C(8,1)] -凤姐虎弟- ♀ (0 bytes) () 10/16/2011 postreply 20:29:22

• 思路please:-) -ttmum- ♀ (0 bytes) () 10/16/2011 postreply 20:49:52