In an exam, a student is required to answer 10 out of 13 questions. Find the number of possible choices if the student must answer: (a) the first two questions; (b) the first or second question, but not both; (c) exactly 3 out of the first 5 questions; (d) at least 3 out of the first 5 questions.

(a) Must answer first 2 questions, thus he needs to answer 8 of the remaining 11 questions.

$\implies C(11,8)=165$ ways. (Answer)

(b) First or second means 2 ways, then he needs to answer 9 of the remaining 11 questions (not both)

$\implies C(2,1)*C(11,9)=110$ ways. (Answer)

(c) Exactly 3 of the first, then he can choose 7 of the remaining 8 questions

$\implies C(5,3)*C(8,7)=80$ ways. (Answer)

(d) Atleast 3 of the first five means, three cases can occur:

CASE 1: 3 of the first five questions $\implies C(5,3)*C(8,7)=80$

CASE 2: 4 of the first five questions $\implies C(5,4)*C(8,6)=140$

CASE 3: 5 of the first five questions $\implies C(5,5)*C(8,5)=56$

Thus, total number of ways $=80+140+56=276$ ways. (Answer)