OA has to be 0 buddy..
Logical explanation: let say all 30 students are a1,a2,a3,...,a30
now a1,a2,a3,...a15 got the recommendation from the 1st prof.
now 2nd prof gives that to 17 student.so take a16,a17,....,a30 which will give u 15 and let say to a1,a2..so total 17.
till now a1,a2 gets both reco from 2 profs rite??
now 3rd prof gives 20 reco...take any 20 out of 28(except a1,a2 as they already got 2 before)..
so it is not necessary they there will be some ppl will get reco from all 3...so ans is 0 as we are looking for least no of ppl.
Now mathematical explanation:
let say the seven sections are a,b,c,d,e,f,g
a=only from prof 1
b=only from prof 2
c=only from prof 3
d=both 1,2
e=both 2,3
f=both 1,3
g=all 1,2,3
now a+b+c+...+f+g=30...
a+b+c+2(d+e+f)+3g=52
d+e+f+2g=22...eq 1
now d+g>=(15+17)-30=2
e+g>=(17+20)-30=7
f+g>=(20+15)-30=5
so d+e+f+3g>=14 and d+e+f+2g=22...2 equations
now 22+g>=14
g>=-8....mathematically only, and again logically not possible
so min value of g=0..
If there is any doubt,please post...i believe for this qn logical approach is quite easy...but for understanding we can have mathematical approach for homework pupose.
