Question

A test consists of 10 multiple choice questions. In how many ways can the test be answered if

(i) Each question has four choices?

(ii) The first four questions have three choices and the remaining have five choices?

(iii) Question number n has n + 1 choices?

Answer 1

(i) Each question has 4 choices. So each questions can be answered in 4 ways. 

Number of Questions = 10 

So they can be answered in 410 ways

(ii) The first four questions have 3 choices. So they can be answered in 3⁴ ways. Remaining 6 questions have 5 choices. So they can be answered in 5⁶ ways. 

So all 10 questions can be answered in 3⁴ × 5⁶

(iii) Given question n has n + 1 choices 

Question 1 has 1 + 1 = 2 choices 

Question 2 has 2 + 1 = 3 choices 

Question 3 has 3 + 1 = 4 choices 

Question 4 has 4 + 1 = 5 choices 

Question 5 has 5 + 1 = 6 choices 

Question 6 has 6 + 1 = 7 choices 

Question 7 has 7 + 1 = 8 choices 

Question 8 has 8 + 1 = 9 choices 

Question 9 has 9 + 1 = 10 choices 

So the number of ways of answering all the 10 questions 

= 2 × 3 × 4 ×…. × 11 = 11! ways