An Easy Problem that is My Favourite

Here's a Python solution that uses strings, such as "lluluulu" or "ullluluu", to represent when Bobby-Joe Bob XXVII goes left, or if he goes up. Bobby-Joe is kept on the plane by limiting the number times he can go left or up to 4. The number of possible ways Bobby-Joe can go will be represented by the number of generated strings of length 8.

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bound = 4 # width of plane
class Directions():
    def __init__(self, steps):
        self.steps = steps
    def adj(self):
        if self.steps.count('l') < bound:
            yield Directions(self.steps+'l')
        if self.steps.count('u') < bound:
            yield Directions(self.steps+'u')
level = {0:[Directions("")]}
cur_lvl = 1
limit = bound*2
while cur_lvl <= limit:
    level[cur_lvl] = set()
    for directions in level[cur_lvl-1]:
        for new in directions.adj():
            level[cur_lvl].add(new)
    del level[cur_lvl-1]
    cur_lvl += 1
print "Answer:", len(level[limit])

Total ways $=^{4+4}C_4=\boxed{70}$