Difference between revisions of "2004 AIME I Problems/Problem 6"
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An integer is called snakelike if its decimal representation <math> a_1a_2a_3\cdots a_k </math> satisfies <math> a_i<a_{i+1} </math> if <math> i </math> is [[odd integer | odd]] and <math> a_i>a_{i+1} </math> if <math> i </math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 have four distinct digits? | An integer is called snakelike if its decimal representation <math> a_1a_2a_3\cdots a_k </math> satisfies <math> a_i<a_{i+1} </math> if <math> i </math> is [[odd integer | odd]] and <math> a_i>a_{i+1} </math> if <math> i </math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 have four distinct digits? | ||
| − | == Solution == | + | == Solution 1 == |
| + | |||
We divide the problem into two cases: one in which zero is one of the digits and one in which it is not. In the latter case, suppose we pick digits | We divide the problem into two cases: one in which zero is one of the digits and one in which it is not. In the latter case, suppose we pick digits | ||
<math>x_1,x_2,x_3,x_4</math> such that <math>x_1<x_2<x_3<x_4</math>. There are five arrangements of these digits that satisfy the condition of being snakelike: <math>x_1x_3x_2x_4</math>, <math>x_1x_4x_2x_3</math>, <math>x_2x_3x_1x_4</math>, <math>x_2x_4x_1x_3</math>, <math>x_3x_4x_1x_2</math>. Thus there are <math>5\cdot {9\choose 4}=630</math> snakelike numbers which do not contain the digit zero. | <math>x_1,x_2,x_3,x_4</math> such that <math>x_1<x_2<x_3<x_4</math>. There are five arrangements of these digits that satisfy the condition of being snakelike: <math>x_1x_3x_2x_4</math>, <math>x_1x_4x_2x_3</math>, <math>x_2x_3x_1x_4</math>, <math>x_2x_4x_1x_3</math>, <math>x_3x_4x_1x_2</math>. Thus there are <math>5\cdot {9\choose 4}=630</math> snakelike numbers which do not contain the digit zero. | ||
In the second case we choose zero and three other digits such that <math>0<x_2<x_3<x_4</math>. There are three arrangements of these digits that satisfy the condition of being snakelike: <math>x_2x_30x_4</math>, <math>x_2x_40x_3</math>, <math>x_3x_40x_2</math>. Because we know that zero is a digit, there are <math>3\cdot{9\choose 3}=252</math> snakelike numbers which contain the digit zero. Thus there are <math>630+252=\boxed{882}</math> snakelike numbers. | In the second case we choose zero and three other digits such that <math>0<x_2<x_3<x_4</math>. There are three arrangements of these digits that satisfy the condition of being snakelike: <math>x_2x_30x_4</math>, <math>x_2x_40x_3</math>, <math>x_3x_40x_2</math>. Because we know that zero is a digit, there are <math>3\cdot{9\choose 3}=252</math> snakelike numbers which contain the digit zero. Thus there are <math>630+252=\boxed{882}</math> snakelike numbers. | ||
| + | |||
| + | == Solution 2 == | ||
| + | |||
| + | *Typing up now -- delete this if solution not typed up by May 26 2017* | ||
== See also == | == See also == | ||
{{AIME box|year=2004|n=I|num-b=5|num-a=7}} | {{AIME box|year=2004|n=I|num-b=5|num-a=7}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 19:00, 25 May 2017
Contents
Problem
An integer is called snakelike if its decimal representation 
satisfies 
if 
is odd and 
if is even. How many snakelike integers between 1000 and 9999 have four distinct digits?
Solution 1
We divide the problem into two cases: one in which zero is one of the digits and one in which it is not. In the latter case, suppose we pick digits
such that 
. There are five arrangements of these digits that satisfy the condition of being snakelike: 
, 
, 
, 
, 
. Thus there are 
snakelike numbers which do not contain the digit zero.
In the second case we choose zero and three other digits such that 
. There are three arrangements of these digits that satisfy the condition of being snakelike: 
, 
, 
. Because we know that zero is a digit, there are 
snakelike numbers which contain the digit zero. Thus there are 
snakelike numbers.
Solution 2
- Typing up now -- delete this if solution not typed up by May 26 2017*
See also
| 2004 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
