In a B-School there are three levels of faculty positions i.e. Professor, Associate Professor and Assistant Professor. It is found that the sum of the ages of all faculty present is 2160, their average age is 36; the average age of the Professor and Associate Professor is 39; of the Associate Professor and Assistant Professor is $$32\frac{8}{11}$$; of the Professor and Assistant Professor is $$36\frac{2}{3}$$; Had each professor been 1 year older, each Associate Professor 6 years older, and each Assistant Professor 7 years older, then their average age would increase by 5 years. What will be the number of faculty at each level and their average ages?

Let the number of professors, associate professors and assistant professors be x, y and z respectively and their average ages be a, b and c respectively.
xa + yb + zc = 2160   -------------------(1)

Average age =36

$$\therefore$$ x+y+z=60

xa + yb = 39 (x + y)
11(yb + zc) = 360(y + z)
3(xa + zc) = 110(x + z)
x(a + 1) + y(b + 6) + z(c + 7) = 2160 + 5*(60) -------(2)            [$$\because$$ Average increases by 5]

Eq 2- Eq 1

We get x+6y+7z=300

Lets solve the options one by one
Option A: x = 16 , y = 24, z = 20

16+6*24+7*20=300 which satisfies the equations.

So either A or C can be the answer.

Now check for the values of a, b, c

a=45, b=35, c=30

ax+by+cz = 45*16+35*24+30*20

=2160

Hence, option A is the correct answer.