Paths to the hill

From a car park it is possible to ascend to the top of a hill on three different hiking trails or by going up by cable car, and in the same four ways it is possible to descend from the hill back to the car park, as illustrated in the picture. By a path to the top of the hill and back the ascent and descent together are meant. Two paths to the top of the hill and back are different, if they differ in the way of ascent or descent.

a) Determine the number of different paths to the top of the hill and back.
b) Determine the number of different paths to the top of the hill and back such that the path from the hill is different from the path to the hill.
c) Determine the number of different paths to the top of the hill and back such that the cable car is used at least once.
d) Determine the number of different paths to the top of the hill and back such that the cable car is used exactly once.

Final Answer:

a =  16
b =  12
c =  7
d =  6

Step-by-step explanation:

$a=4\cdot 4=16$

$b=4\cdot 3=12$

$c=3\cdot 1+1\cdot 3+1\cdot 1=7$

$d=3\cdot 1+1\cdot 3=6$

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