Difference between revisions of "Talk:2021 Fall AMC 12B Problems/Problem 17"
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| + | ==Solution 3== | ||
| + | Use generating function, define <math>c_{n}\cdot x^{n}</math> be <math>c_{n}</math> ways for the end point be <math>{n}</math> unit away from the origins. | ||
| + | |||
| + | Therefore, | ||
| + | |||
| + | if the current point is origin, need to <math>\cdot6{x}</math> | ||
| + | |||
| + | if the current point on vertex of the unit hexagon, need to <math>\cdot(x^{-1}+2)</math>, where there is one way to return to the origin and there are two ways to keep distance = 1 | ||
| + | |||
| + | Now let's start with <math>p(x)=1</math>; | ||
| + | |||
| + | 1st step: <math>p(x)=6x</math> | ||
| + | |||
| + | 2nd step: <math>p(x)=6x\cdot(x^{-1}+2) = 6 + 12x </math> | ||
| + | |||
| + | 3rd step: <math>p(x)=6\cdot6x + 12x\cdot(x^{-1}+2) = 12 + 60x</math> | ||
| + | |||
| + | 4th step: <math>p(x)=12\cdot6x + 60x\cdot(x^{-1}+2) = 60 + 192x </math> | ||
| + | |||
| + | 5th step: <math>p(x)=60\cdot6x + 192x\cdot(x^{-1}+2) = 192 + 744x </math> | ||
| + | |||
| + | So there are <math>192+744=936</math> ways for the bug never moves more than 1 unit away from orign, and <math>\frac{936}{6^5} = \boxed{\frac{13}{108}}.</math> | ||
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| + | |||
| + | -wwei.yu | ||
| + | |||
| + | == /*Solution 3*/ == | ||
| + | |||
==Solution 3== | ==Solution 3== | ||
Use generating function, define <math>c_{n}\cdot x^{n}</math> be <math>c_{n}</math> ways for the end point be <math>{n}</math> unit away from the origins. | Use generating function, define <math>c_{n}\cdot x^{n}</math> be <math>c_{n}</math> ways for the end point be <math>{n}</math> unit away from the origins. | ||
Revision as of 14:51, 28 November 2021
Solution 3
Use generating function, define 
be 
ways for the end point be 
unit away from the origins.
Therefore,
if the current point is origin, need to 
if the current point on vertex of the unit hexagon, need to 
, where there is one way to return to the origin and there are two ways to keep distance = 1
Now let's start with 
;
1st step: 
2nd step: 
3rd step: 
4th step: 
5th step: 
So there are 
ways for the bug never moves more than 1 unit away from orign, and 
-wwei.yu
/*Solution 3*/
Solution 3
Use generating function, define be ways for the end point be unit away from the origins.
Therefore,
if the current point is origin, need to
if the current point on vertex of the unit hexagon, need to , where there is one way to return to the origin and there are two ways to keep distance = 1
Now let's start with ;
1st step:
2nd step:
3rd step:
4th step:
5th step:
So there are ways for the bug never moves more than 1 unit away from orign, and
-wwei.yu
