SOLUTION: The University of Ballygobackwards has three colleges: - the College of Science, which has 19 staff, 5 of whom are female; - the College of Engineering, which has 6 staff, 4 of w

Question 1176466: The University of Ballygobackwards has three colleges:
- the College of Science, which has 19 staff, 5 of whom are female;
- the College of Engineering, which has 6 staff, 4 of whom are female;
- and the College of Arts, which has 18 staff, 8 of whom are female.
Each college has a Dean, who is chosen from the staff in that college.
a)What is the total number of ways these three Deans can be chosen?
b)How many ways can three Deans be chosen, so that at least one of them is female?
Answer by ikleyn(52909)   (Show Source): You can put this solution on YOUR website!
.
The University of Ballygobackwards has three colleges:
- the College of Science, which has 19 staff, 5 of whom are female;
- the College of Engineering, which has 6 staff, 4 of whom are female;
- and the College of Arts, which has 18 staff, 8 of whom are female.
Each college has a Dean, who is chosen from the staff in that college.
a) What is the total number of ways these three Deans can be chosen?
b) How many ways can three Deans be chosen, so that at least one of them is female?
~~~~~~~~~~~~~

            The post contains many words and seems to be a complicated problem, from the first glance.

            But actually, it is EXTREMELY simple.

(a)  For the 1st college, there are 19 choices;

     For the 2nd college, there are  6 choices;

     For the 3rd college, there are 18 choices.


     These choices are independent, since we choose in different / (separate) sets.


     THEREFORE, the total number of triples of the three Deans is the product  19*6*18 = 2052.     ANSWER


         (according to the Fundamental counting principle).



(b) In this part, we should subtract from the total number of cases 2052 of the part (a)

    the number of cases  (19-5)*(6-4)*(18-8) = 13*2*10 = 260, when all the three Deans are men.


    So, the answer is  2052 - 260 = 1792.

Solved, answered, explained and completed.

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