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2004 AIME I Problems/Problem 6

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Problem

An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ 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 $x_1,x_2,x_3,x_4$ such that $x_1<x_2<x_3<x_4$. There are five arrangements of these digits that satisfy the condition of being snakelike: $x_1x_3x_2x_4$, $x_1x_4x_2x_3$, $x_2x_3x_1x_4$, $x_2x_4x_1x_3$, $x_3x_4x_1x_2$. Thus there are $5\cdot {9\choose 4}=630$ snakelike numbers which do not contain the digit zero.

In the second case we choose zero and three other digits such that $0<x_2<x_3<x_4$. There are three arrangements of these digits that satisfy the condition of being snakelike: $x_2x_30x_4$, $x_2x_40x_3$, $x_3x_40x_2$. Because we know that zero is a digit, there are $3\cdot{9\choose 3}=252$ snakelike numbers which contain the digit zero. Thus there are $630+252=882$ snakelike numbers.

See also

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