# Difference between revisions of "2004 AIME I Problems/Problem 6"

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== Solution == | == Solution == | ||

− | {{ | + | 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. | ||

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+ | 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=882</math> snakelike numbers. | ||

== See also == | == See also == | ||

* [[2004 AIME I Problems/Problem 5| Previous problem]] | * [[2004 AIME I Problems/Problem 5| Previous problem]] |

## Revision as of 14:36, 29 November 2006

## 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

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.