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A haiku is a poem with three lines: the first line contains  syllables, the second line  syllables, and the last line  syllables. If each word in each list shown is used at most once, how many different haiku can be made with these words?

\(\begin{array}{c|c} \textbf{2-syllable} & \textbf{3-syllable} \\ \hline \textbf{UNKNOWN} & \textbf{ALGEBRA} \\ \text{MEASURE} & \text{TRIANGLE} \\ \text{COUNTING} & \text{REASONING} \\ \text{LOGIC} & \end{array}\)

 Jun 16, 2023
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To count the number of haiku, we can first count the number of ways to choose the words for each line. The first line can have any of the 8 words, the second line can have any of the 7 words, and the last line can have any of the 8 words. This gives us 8×7×8=448 possible combinations of words.

However, not all of these combinations are valid haiku. For example, a haiku cannot have two words of the same length in the same line. This means that we need to remove some of the combinations.

There are 4 ways to choose 3 words of the same length for the first line. There are 3 ways to choose 2 words of the same length for the second line. And there are 4 ways to choose 3 words of the same length for the last line. This gives us 4+3+4=11 combinations of words that are not valid haiku.

Subtracting these 11 combinations from the 448 possible combinations gives us 437 valid haiku.

 Jun 16, 2023

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