Answer to Question #98514 in Discrete Mathematics for Ahmed

Question #98514
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.
1
Expert's answer
2019-11-15T11:48:46-0500

(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)

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