Help the grasshopper
A 10 X 10 X10 cube is formed of small unit cubes. A grasshopper sits in the center O of one of the corner cubes. At a given moment it can jump to the center of any of the cubes which has a common face with the cube where it sits, as long as the jump increases the distance between point O ant the current position of the grasshopper. How many ways are there for grasshopper to reach the unit cube at the opposite corner?
answer is in the form
Find
The answer is 39.
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1 solution
Let's say the grasshopper starts in the front, left, uppermost cube. (We can always rotate the cube so that this is true). He must move nine cubes right, nine cubes down, and nine cubes towards the back of the large cube. He cannot move left, towards the front or top of the cube without approaching O. Therefore he makes 27 steps. These 27 steps can happen in 27! different orders, but the nine moves forward are counted as identical to each other, as are the nine moves right and down. This means that we have counted each arrangement of these individual sets of moves 9! times. Therefore the number of orders which we can carry out the 27 steps in is actually 27!/(9! 9! 9!)=27/(9!^3). 27+9+3=39.