model these models

[Shreya Agrawal]

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?

[Shreya Agrawal]

PLEASE someone answer!

[Amit Goyal]

For simplicity, Let me name the models a, b, c, d, e, f, and g in decreasing order of their heights. So, we want five models to stand in increasing order of their heights, and forth-tallest i.e. d and sixth-tallest i.e.  f must not stand adjacent to each other.
We will count it by first finding total number of arrangements of five models where models stand in increasing order of their heights and subtracting from it the arrangements in which five models stand in increasing order of their heights and d and f are selected but not e.
C(7, 5) -  C(4, 3) = 7!/(5!2!)  - 4!/(3!1!) = 21 - 4 = 17

[Shreya Agrawal]

Thank you sir.