Lattice Path Consider Lattice Paths With North East And South East Steps

1. Lattice path: consider lattice paths with North-east and South-east steps. how many lattice paths are there from the point of coordinates (0,0) to the point of coordinates (0,b) where b is a non negative integer.
2. Suppose you are a biologist, you sequence a gene whose DNA sequence has length 21 and you observe that everytime the nucleotide A appears, it is followed by the nucleotide C. An immediate question is to wonder if that is something we can expect from a random sequence or if this might be exceptional enough that the biological function of this observation is investigated. To answer this we need to find the number of strings of size 21 over the alphabet {A,C,G,T} in which an A is always followed by a C.