[8] A caterpillar encounters a 3-dimensional lattice resting on the ground, having dimensions 5" x 4" x 3". The width of the lattice, running east-west is x = 5", the length, running north- south is y = 4" and the height, running up-down is z = 3". One corner of the lattice is at the point (x, y, z) = (0, 0, 0) and the far corner, in an east, north, and up direction is (5, 4, 3). The caterpillar encounters the lattice at (0, 0, 0) and will finish at (5, 4, 3). Each node the caterpillar chooses will move it closer to (5, 4, 3). The caterpillar will reach the endpoint in 12 moves traveling only east, north, or up. If the caterpillar always randomly chooses the next lattice point toward its goal, [a] How many unique paths can it take to reach the goal? [b] What is the probability that the caterpillar passes through (2, 1, 2) on the way to (5, 4, 3)