The probability that Mr $A$ passed his friends house is Mr $A$ lives at origin on the cartesian plane and has his office at $(4,5).$His friend lives at $(2,3)$ on the same plane.Mr $A$ can go to his office travelling one block at a time either in the +y or +x direction.If all possible paths are equally likely,then the probability that Mr $A$ passed his friends house is
$(A)\frac{1}{2}\hspace{1cm}(B)\frac{10}{21}\hspace{1cm}(C)\frac{1}{4}\hspace{1cm}(D)\frac{11}{21}$
I could not start this question,i tried hard.Please help me.
 A: Just one more perspective to look at the same solution as zoli's proposed.


*

*To reach $(4,5)$ from $(0,0)$, we have to use $4$ moves in $+x$ and $5$ moves in $+y$ direction. So any path that we chose will be one of the permutation of $xxxxyyyyy$. 
So we have  $\dfrac{9!}{(4!\times5!)} = 126$ ways.

*Similarly the number of ways to reach from $(0,0)$ to $(2,3)$ (i.e.$ xxyyy$) and from $(2,3)$ to $(4,5)$ (i.e. $xxyy$) can be computed. The total number of ways in which we can reach from $(0,0)$ to $(4,5)$ via $(2,3)$ is product of these two events(as the event are happening one after the other). So we have $\dfrac{5!}{2!*3!} \times \dfrac{4!}{2!\times 2!} = 10 \times 6 = 60$ possible paths.     
Total number of possible events (Sample space) =$ 126$
Number of events of interest = $60$
As the probability of every path is equally likely, we can divide them to get the required probability.
$ \dfrac{60}{126} = \dfrac{10}{21}$
A: The right answer is B ; $$\frac {10}{21}$$.
To reach the point $(4,5)$ you have to take $9$ steps. Out of these $9$ steps you can choose $4$ to be taken to the right. As a total you have $${9\choose 4}=126$$ possibilities.
To reach the point $(2,3)$ you have $5$ steps out of which you can choose $2$ to be taken to the right. You have then ${5\choose 2}=10$ choices. From $(2,3)$ there are $4$ steps to $(4,5)$ out of which steps you can choose $2$ to be taken to the right. To reach you office through $(2,3)$ you have 
$${5\choose 2}\times {4\choose 2}=60$$
possibilities.
So, the probability sought is
$$\frac{60}{126}=\frac{10}{21}.$$
