How many ways are there to arrive at point $E$ in $10$ minutes or less? Otto is traveling along octagon $ABCDEFGH$. He starts at $A$ and every minute, he will move to one of the adjacent vertices. He will stop once he reaches $E$. How many ways are there to arrive at point $E$ in $10$ minutes or less?


This question seems hard as time is the constraint here. How do I account for the time constraint here? my first approach would be to do case work with the amount of minutes: in $4$ minutes there are two possibilities, it is impossible for $5$ minutes, and then the casework gets more tricky.
You could draw something similar to Pascal's triangle where each cell is the sum of the two cells diagonally above it (except in columns D and F since you cannot come from E).

It might look something like this, and you then have to add up the numbers in the two E columns.

E   F   G   H   A   B   C   D   E
0   0   0   0   1   0   0   0   0
0   0   0   1   0   1   0   0   0
0   0   1   0   2   0   1   0   0
0   1   0   3   0   3   0   1   0
1   0   4   0   6   0   4   0   1
0   4   0   10  0   10  0   4   0
4   0   14  0   20  0   14  0   4
0   14  0   34  0   34  0   14  0
14  0   48  0   68  0   48  0   14
0   48  0   116 0   116 0   48  0
48  0   164 0   232 0   164 0   48


Apparently I calculated this in 2001
To go forward 4 places he must have 7 steps forward and 3 backward. He must make 10 steps so the number of ways is ${10 \choose 7} = 120$. He could also get there by 7 steps backward and 3 forward so double this to get $240$.