1. There are 41 candidates approved to be allotted as the F.D.R.E. council of cabinets. Historic facts of the candidates are shown below, 8 were served as former cabinet members,14 were university presidents, 15 served as members of P.R.H., 2 served in former cabinet and as P.R.H, 4 served as university presidents and members of former cabinet, 6 were served as university presidents and members of P.R.H, 1 served in all the three positions. The find the number of candidates: a. Served in none of the three positions? b. Only university presidents? c. At least one of the three positions? d. Exactly two positions?

a.

$N(C\cup Pr\cup P.R.H)=N(C)+N(Pr)+N(P.R.H)$

$-N(C\cap P.R.H)-N(C\cap Pr)-N(Pr\cap P.R.H)$

$+N(C\cap Pr\cap P.R.H)$

$=8+14+15-2-4-6+1=26$

$N((C\cup Pr\cup P.R.H)^C)=41-N(C\cup Pr\cup P.R.H)$

$=41-26=15$

15 candidates Served in none of the three positions.

b.

$N(only\ Pr)=N(Pr)-N(C\cap Pr)$

$-N(Pr\cap P.R.H)+N(C\cap Pr\cap P.R.H)$

$14-4-6+1=5$

5 candidates served only university presidents.

c. 

$N(C\cup Pr\cup P.R.H)= 26$

26 candidates served at least one of the three positions.   

d.   

$N(exactly\ two\ positiions)$

$=N(C\cap P.R.H)-N(C\cap Pr\cap P.R.H)$

$+N(C\cap Pr)-N(C\cap Pr\cap P.R.H)$

$+N(Pr\cap P.R.H)-N(C\cap Pr\cap P.R.H)$

$=2-1+4-1+6-1=9$

9 candidates served exactly two positions.