Willy Walks to School (Part I)

Willy lives in the Cartesian coordinate plane. Willy's house lies at ( 0 , 0 ) (0,0) , while his school, Takoma Park, lies at ( 10 , 12 ) (10,12) . Willy needs to walk to school in the shortest way possible, or else he will be late to school and receive three warnings from Mr. Siddique. How many ways are there to get from the origin to ( 10 , 12 ) (10,12) moving only up and to the right?

(Note that Willy must walk on the roads, or parallel to the x x - or y y -axis. If he is caught not doing so, he will receive three warnings.)


The answer is 646646.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Let us think of each movement through the roads as a pace . That is, the movement from ( x , y ) (x, y) to ( x + 1 , y ) (x + 1, y) or from ( x , y ) (x, y) to ( x , y + 1 ) (x, y + 1) is one pace. Since it takes 10 10 paces to travel from the origin to ( 10 , 0 ) (10, 0) and 12 12 paces to travel from the origin to ( 0 , 12 ) (0, 12) , travelling to ( 10 , 12 ) (10, 12) takes 22 22 paces.

Considering that Willy is to walk strictly either right or up to go to school in the most efficient way, then, we can think of those twenty-two paces as a combination of binary movements that either go up or right. Now, if Willy has to advance strictly ten paces to the right (and the rest are 'up' paces), there are ( 22 10 ) = 646646 {22 \choose 10} = 646 646 ways to do so. ( NOTE : You will arrive at the same result if you considered the twelve 'up' paces instead of the ten 'right' paces: ( 22 12 ) = 646646 {22 \choose 12} = 646 646 .)

wow!! brilliant

Sahil Gohan - 7 years, 1 month ago

Is there a simpler way to compute the result? The logic was trivial. Thanks

Aravind Narayanan - 2 years ago
Tanya Gupta
Mar 29, 2014

Hey William Sorry to interrupt in another question but I think your answer for willy ....part 2 is wrong... Thinking in the same way...answer should be 12C6*3C2....is something wrong in here????

Oops.

William Cui - 7 years, 1 month ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...