Given : Number of individual needed to be selected to form jury = 4
Number of students = 13
Number of faculty members = 6
Total = 13+6=19
Using combinations , the number of combinations of choosing jury of 4 individuals
= [tex]^{19}C_{4}=\dfrac{19!}{4!(19-4)!}\\\\=\dfrac{19\times18\times17\times16\times15!}{4!15!}=3876[/tex]
a) Number of ways of selecting jury of all students
= [tex]^{13}C_{4}=\dfrac{13!}{4!(13-4)!}\\\\=\dfrac{13\times12\times11\times10\times9!}{4!9!}=715[/tex]
The probability of selecting a jury of all faculty=[tex]\dfrac{715}{3876}=0.1845[/tex]
b) The number of ways of selecting jury of all faculty=[tex]^{6}C_{4}=\dfrac{6!}{4!(6-4)!}=15[/tex]
The probability of selecting a jury of all faculty=[tex]\dfrac{15}{3876}=0.0039[/tex]
c) The number of ways of selecting jury of 2 students and two faculty:
[tex]^{13}C_{2}\times ^{6}C_2\\\\=\dfrac{13!}{2!(13-2)!}\times\dfrac{6!}{2!(6-2)!}\\\\=1170[/tex]
Now, the probability of selecting a jury of six students and two two faculty
[tex]\dfrac{1170}{3876}=0.3019[/tex]
