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?

Total = A + B + C + D

where A = Number of arrangements that include 4th but not 6th

B = Number of arrangements that include 6th but not 4th

C = Number of arrangements that include both 4th and 6th

D = Number of arrangements that include neither

A = 5C4 x 1 = 5

B = 5C4 x 1 = 5

C = 5C3 x 1 = 10

D = 5C5 x 1 = 1

Total = 21

Lemme explain one term:

A = 5C4 x 1

5C4 = number of ways of choosing the members of the arrangement.

1 = Number of ways of arranging the 5 chosen models

In A, one member(4th) is already picked and one member(6th) is dropped. We just need to choose 4 more members from the remaining 5. Hence number of ways of choosing members for A is 5C4.

There is only 1 way to arrange the 5 chosen models.

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