081 Graph Count number of paths between two nodes Theory

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Hey nice explaination !!
But can you tell me that can we do this without using visited array bcz what we are doing is first mark he node and then unmark it ... So, isn't it possible that we could do without marking the nodes ???
I mean where is the need to first mark the node and then unmark it again. Won't it be simple if we could just solve it without using mark and then unmark concept ???

prateeksomani