Counting Routes!

Each number on the grid represents the number of routes to that node (intersection point on the grid).

For example, consider the node which has $15$ routes to it on the bottom row.

As there are $10$ routes to the node above it and $5$ routes to the node on its the right , there are $10 + 5 = 15$ routes to it in total.

In the same way, each of the other values have been calculated and it becomes apparent that the total number of different routes through a $4 \times 4 ,grid ,is, \boxed{70}$ .

$Generalization$ - Observe these using the Pascal's triangle. The number of routes are circled with red color.

Routes for $1 \times 1$ grid $= {2 \choose 1} = 2$ .

Routes for $2 \times 2$ grid $= {4 \choose 2} = 6$ .

Routes for $3 \times 3$ grid $= {6 \choose 3} = 20$ .

Routes for $4 \times 4$ grid $= {8 \choose 4} = 70$ .

$\dots$

$\dots$

Routes for $n \times n$ grid = $\Large {2n \choose n} = \frac{(2n)!}{n! \times n!}$ :)

The routes can be represented by $4 \ R$ 's and $4 \ D's$ [R=Right, D=Down]. Now, the number of routes is equal to the number of permutations of $4 \ R$ 's and $4 \ D's$ . So thee answer is $\dbinom{8}{4}=\fbox{70}$