Question

Simple with work shown. Thank you!

1. n 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.

Answer 1

Number of questions available to answer =13

Number of questions the student is required to answer = 10

(a)

The student must answer first two questions; Therefore number questions available = 13-2=11

Remaining questions to be answered = 10-2 =8

Number of possible choices of choosing 8 questions from 11 questions = =165

(b) the student must answer

The First or Second question , but not both

Case 1 : Students must answer Question 1 (and not answer question 2) :  

Therefore number questions available = 13-2=11

Remaining questions to be answered = 10-2 =8

Number of possible choices of choosing 8 questions from 11 questions = =165

Case 2 : Students must answer Question 2 (and not answer question 1) :  

Therefore number questions available = 13-2=11

Remaining questions to be answered = 10-2 =8

Number of possible choices of choosing 8 questions from 11 questions = =165

Number of possible choices = 165+165 =330

(c)

The student must answer exactly 3 out of the first 5 questions;

Number of possible choices of answering exactly 3 out of the first 5 questions = =10

Number of questions available to answer = 13-5 =8

Remaining questions to be answered = 10-3 =7

Number of possible choices of answering 7 of the 8 questions = = 8

Total number of possible choices = 10 x 8 = 80

(d)

The student must answer at least 3 out of the 5 questions

Case 1 : The student must answer exactly 3 out of the first 5 questions; From (c)

Number of choices = 80

Case 2 : The student must answer exactly 4 out of the first 5 questions;

Number of choices of answering exactly 4 out of the first 5 questions = =5

Number of questions available to answer = 13-5 =8

Remaining questions to be answered = 10-4 =6

Number of choices of answering 6 of the 8 questions = = 28

Total number of choices = 5 x 28 = 140

Case3 : The student must answer exactly 5 out of the first 5 questions;

Number of choices of answering exactly 5 out of the first 5 questions = 1

Number of questions available to answer = 13-5 =8

Remaining questions to be answered = 10-5 =5

Number of choices of answering 6 of the 8 questions = = 56

Total number of choices = 1 x 56 = 56

Total number of choices

= Number of choices in case 1 + Number of choices in case 2 + number of choices in case 3 = 80+140+56 = 276

Number of possible choices = 276