Note that all of these codes are different,and amongst them there is exactly one,namely 00110011,which describes a valid mountain profile,In fact, this always happens,whichever of the 70 codes is chosen as the starting point.It follows that the 70 codes have been divided into 14 classes each consisting of 5 code and each containing one valid code. Hence there are 14 valid codes and the Catalan formula (1) is demonstrated.
It is now time to prove the two claims made above.