Out of seven models, all of different heights, five models will be chosen to pose for a photograph. If the five models are to stand in a line from shortest to tallest, and the fourth-tallest and sixth-tallest models cannot be adjacent, how many different arrangements of five models are possible?

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Correct

210

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Official Explanation

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Notice that since the fourth-tallest and sixth-tallest models cannot be adjacent and models are to stand in a line from shortest to tallest, then if we choose 4th and 6th we MUST also choose 5th to stand between them.

Also notice that since the models are to stand in a line from shortest to tallest then for any group of 5 models we choose, there will be only one arrangement possible: from shortest to tallest.…

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What this question tests

What it tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Common trap

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.