.
In a certain city, there are seven streets going north-south and five streets going east-west.
How many street paths start at the southwest corner of the city, end at the northeast corner of the city,
and have the shortest possible length?
~~~~~~~~~~~~~~~~~
Each such path is a combination of elementary 7 south->north parts/blocs and
5 west->east parts/blocks, 7+5 = 12 blocks, in total.
So, using designation/letter V for south->north blocks ("vertical") and designation/letter H ("horizontal")
for west->east blocks, each path can be coded as a word of 12 letters "V" and "H".
There is one-to-one correspondence between such paths and such words.
So, we can say that the number of paths equals the number of words of the length 12,
written using letter V and H. Repeating of letters is allowed.
From Combinatorics, it is known that the number of such words is
= after reduction of the fraction = = = = 792.
Indeed, from the total 12 positions of the word, 7 positions for letters "V" can be selected
in different ways; the rest 5 positions are occupied by the letters "H".
It gives us the answer to the problem's question: there are 792 such paths.
Solved.
-------------------
The closest combinatorial problem is this:
How many distinguishable ways are there to place in line 7 red and 5 white balls, if they distinct only by color ?"
//////////////////
To see many other similar (and different) solved problems, look into the lesson
- Arranging elements of sets containing indistinguishable elements
in this site.