In the picture below, you see a schematic of some of the streets in a certain town. Determine how many paths exist from $A$ to $B$ that travel only to the right and up. Two such paths are given in the figure, one using a dashed line and one using a dotted line (they overlap in multiple places).
So basically, you can only travel up and right only.
So what I decided to do was create some dots.
Basically, I said the number of paths(P) is equal to the combined totals of path going through each point.
So :
P=(# of paths from A $\rightarrow$ V $\rightarrow$ B)+(# of paths from A $\rightarrow$ W $\rightarrow$ B)+(# of paths from A $\rightarrow$ X $\rightarrow$ B)+(# of paths from A $\rightarrow$ Y $\rightarrow$ B)+(# of paths from A $\rightarrow$ Z $\rightarrow$ B)
Then, I started calculating each part individually. I said (# of paths from A $\rightarrow$ V $\rightarrow$ B)=$15$
and (# of paths from A $\rightarrow$ W $\rightarrow$ B)=$30$
but then I realized that these numbers are too big and I must doing something wrong. If anyone wants to know how I am calculating, this is how:
(# of paths from A $\rightarrow$ W $\rightarrow$ B)=(# of paths from A $\rightarrow$ W) *(# of paths from W $\rightarrow$ B )
Any help would be most appreciated.
Hint:
$8$ moves have to be made. $4$ up and $4$ to the right. So you must select $4$ out of $8$ to be the upward moves. The other moves are automatically the moves to the right.
The two examples are $RURRURUU$ (dashed) and $RRUUURRU$ (dotted).
So it comes to calculating the number of ways $4$ items can be selected from $8$.