Photo line with heights falling away from the middle

Question

Ten students, all of different heights, line up in a row of ten for a photo. The two tallest must occupy the two middle spots, and moving outward from the middle in either direction the heights must strictly decrease. How many lineups are possible?

Short answer

$140$ . The two tallest can swap places in $2$ ways. Of the other eight, choose which four stand on the left; each side then has exactly one valid decreasing order. Total $2\binom84=140$ .

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Lessons behind this question

Counting and Combinatorics

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