Willy lives in the Cartesian coordinate plane. Willy's house lies at , while his school, Takoma Park, lies at . 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 moving only up and to the right?
(Note that Willy must walk on the roads, or parallel to the - or -axis. If he is caught not doing so, he will receive three warnings.)
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Let us think of each movement through the roads as a pace . That is, the movement from ( x , y ) to ( x + 1 , y ) or from ( x , y ) to ( x , y + 1 ) is one pace. Since it takes 1 0 paces to travel from the origin to ( 1 0 , 0 ) and 1 2 paces to travel from the origin to ( 0 , 1 2 ) , travelling to ( 1 0 , 1 2 ) takes 2 2 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 ( 1 0 2 2 ) = 6 4 6 6 4 6 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: ( 1 2 2 2 ) = 6 4 6 6 4 6 .)