A photographer will arrange 6 people of 6 different heights for photograph by placing them in two rows of three so that each person in the first row is standing in front of someone in the second row. The heights of the people within each row must increase from left to right, and each person in the second row must be taller than the person standing in front of him or her. How many such arrangements of the 6 people are possible?
5
6
9
24
36
接下来讨论其余四个数的排列:
当第一排第二个数是2时,第一排第三个数在其余三个数中选择一个即可,选完后剩下的两个数按照大小顺序排在第二排剩余位置时只有一种情况。因此第一排第二个数是2时有3中排法。
同理,当第一排第二个数是3时,第一排最后一个数只能在4、5中二选一,选完后剩下的一个和2按照大小排在第二排。第一排第二个数是3时有两种排法。
而当第一排第二个数是4时,第一排第三个数和第二排第二个数都需要比4大,此时只有5比4大,排法不成立。同理,第一排第二个数也不可能是5。
综上所述,总计有3+2=5种排法,分别是:
4,5,6 3,5,6 3,4,6 2,5,6 2,4,6
1,2,3 1,2,4 1,2,5 1,3,4 1,3,5