Note: Please do not mark this question as duplicate of Combination of smartphones' pattern password because no answer given there is correct and I have an extra question related to it.
This is a smartphone Pattern lock.
The rules are as follows:
1. At least 4 nodes should be selected.
2. The pattern should be continuous.
3. Cycles are not allowed.
4. Any node cannot be revisited.
So, if we choose only four nodes. The number of arrangements are $^9P_4$.
If we choose five, the number of arrangements are $^9P_5$ and so on.
But the total number of possible combinations is not $^9P_4 + ^9P_5 + ^9P_6 + ... + ^9P_9$.
The problem is here, that, if we choose an arrangement in which while connecting two points we go over another point, we are counting it twice. For example, if we choose the four point in the corners and join them cyclically, it goes over three more points so it comes under $^9P_4$ and also under $^9P_5$, $^9P_6$ and $^9P_7$.
What shall I do to remove these extra configurations? Is there any other problem with my method?
