From a faculty of six professors, six associate professors, ten assistant professors, and twelve instructors, a committee of size six is formed randomly. What is the probability that

(a) there are exactly two professors on the committee;

(b) all committee members are of the same rank

Answer: a) 0.23

b) 0.00085

Step-by-step explanation:

6 professors

6 associate professors,

10 assistant professors,

12 instructors

Committee: 6 people

Total of combinations possible:

C₃₄,₆ = 34! = 1344904

6! 28!

(a) there are exactly two professors on the committee;

C₆,₂ * C₂₈,₄ = 6! * 28! = 15*20475 = 307125

2! 4! 4! 24!

P = 307125/1344904 = 0.23

(b) all committee members are of the same rank

All prof: C₆,₆ = 1

All assoc. prof: C₆,₆ = 1

All assist prof: C₁₀,₆ = 210

All instr: C₁₂,₆ = 924

1+1+210+924 = 1136

P = 1136/1344904 = 0.00085