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.
Anyone who has never made a mistake has never tried anything new. -Albert Einstein.