LatticeMethod CountTheOpposite AMC8 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?
This problem can be solved in different ways. The first approach is manual counting. Manual counting can be performed more systematically using the lattice method as show below. The answer is $\boxed{4}$.
Alternatively, because there is only one blockage and the overall shape is regular (i.e. rectangle), this problem can also be solved using the counting by the opposite method. If there is no blockage, there are $C_{3+2}^2 = 10$ different ways. Among these different ways, there are $C_{1+1}^1 \times C_{2+1}^1 = 6$ ways will pass the blockage. Therefore, the number of routes without passing the blockage is $10-6 = 4$.