Annette from Copleston Sixth Form explains why there are $56$ possible routes from $A$ to $B$:
All routes between $A$ and $B$ will be equal in length as you need to
travel $3$ east and $5$ north in any order. In total there are $56$ different
paths leading to $B$.
$1$
$6$
$21$
$56$ ($B$)
$1$
$5$
$15$
$35$
$1$
$4$
$10$
$20$
$1$
$3$
$6$
$10$
$1$
$2$
$3$
$4$
$A$
$1$
$1$
$1$
The number in each square represents the number of possible paths from $A$ to that square.
It is worked out like this:
$M$
$M+N$
-
$N$
because you must arrive at a square from the point either directly south or directly west of it.
Leah from St. Stephen's School Carramar found a general rule for the number of ways of travelling from point $(x_1,y_1)$ to $(x_2,y_2)$ going only north and east and discussed some ways to find the number of paths from $A$ to $B$:
If the route must be taken without crossing diagonally, it will take $(x_2-x_1) + (y_2-y_1)$
units to reach the opposite point.
E.g the points $(0,1)$ and $(3,6)$, a path will be $(6-0)+(3-1) = 8$ units long.
Number of paths from $(x_1,y_1)$ to $(x_2,y_2)$ = number of paths consisting of $(x_2-x_1)$ steps east and $(y_2-y_1)$ steps north, in some order