Answer to Question #288249 in Discrete Mathematics for helenabebe

Question #288249

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?


1
Expert's answer
2022-01-19T14:10:47-0500


a.


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

N(CP.R.H)N(CPr)N(PrP.R.H)-N(C\cap P.R.H)-N(C\cap Pr)-N(Pr\cap P.R.H)

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

=8+14+15246+1=26=8+14+15-2-4-6+1=26

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

=4126=15=41-26=15

15 candidates Served in none of the three positions.


b.


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

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

1446+1=514-4-6+1=5

5 candidates served only university presidents.


c. 


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

26 candidates served at least one of the three positions.   


d.   


N(exactly two positiions)N(exactly\ two\ positiions)

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

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

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

=21+41+61=9=2-1+4-1+6-1=9

9 candidates served exactly two positions.



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