Question 2.32: The faculty in an academic department in UC-Davis consists o......
The faculty in an academic department in UC-Davis consists of 4 assistant professors, 6 associate professors, and 5 full professors. Also, it has 30 graduate students. An ad hoc committee of 5 is to be formed to study a certain curricular matter.
(i) What is the number of all possible committees consisting of faculty alone?
(ii) How many committees can be formed if 2 graduate students are to be included and all academic ranks are to be represented?
(iii) If the committee is to be formed at random, what is the probability that the faculty will not be represented?
It is clear that combinations are the appropriate tool here.
Then we have:
(i) This number is: {\binom{15}{5}}={\frac{15!}{5!10!}}={\frac{11\times12\times13\times14\times15}{1\times2\times3\times4\times5}}=3,003.
(ii) Here the number is: {\binom{30}{2}}{\binom{4}{1}}{\binom{6}{1}}{\binom{5}{1}}=\textstyle{\frac{30!}{2!28!}}\times4\times6\times5={\frac{29\times30}{2}}\times120=52,200.
(iii) The required probability is:
{\frac{{\binom{30}{5}}{\binom{15}{0}}}{{\binom{45}{5}}}}={\frac{{\binom{30}{5}}}{{\binom{45}{5}}}}={\frac{30!/5!25!}{45!/5!40!}}={\frac{26\times 27\times 28\times 29\times 30}{41\times42\times43\times44\times45}}={\frac{2,262}{19,393}}\simeq0.117.