I'm having trouble with the following:

You live at the south-west corner of a grid of streets 9 blocks north-to-south and 6 blocks east-to-west. How many possible efficient paths are there to your office at the north-east corner if the east-west block which begins at 3 blocks east and 4 blocks north of your home is closed?

I'm stumped!!

2. Hello bjanela
Originally Posted by bjanela
I'm having trouble with the following:

You live at the south-west corner of a grid of streets 9 blocks north-to-south and 6 blocks east-to-west. How many possible efficient paths are there to your office at the north-east corner if the east-west block which begins at 3 blocks east and 4 blocks north of your home is closed?

I'm stumped!!
Suppose your home is H, your office O and that the block that's closed has A and B at its western and eastern ends, respectively.

With no restrictions on the route, the number of routes from H to O is $\displaystyle {^{15}C_6}=5005$, since this is the number ways in which the positions of the 6 easterly 'legs' of the journey may be positioned within the 15 legs that make up the whole journey.

The number of routes from H to A is similarly $\displaystyle {^7C_3}=35$ and the number of routes from B to O is $\displaystyle {^7C_2}=21$. Thus the number of routes from H to O via A and B is $\displaystyle 35 \times 21 = 735$. These represent the routes that are not available if AB is closed.

So the number of available routes that don't go via A and B $\displaystyle = 5005 - 735 = 4270$.