Question: A question paper consists of 5 problems, each problem having 3 internal choices. In how many ways can a candidate attempt one or more problems?

A: 63
B: 511
C: 1023
D: 31

Ans: C
Solution: There are 5 problems. Each problem having three choices to answer. (Suppose the question as a multiple choice question having 3 options)

So a candidate can either answer 1, 2, 3, 4 or all 5 questions.

No of ways to answer only one question = ⁵C₁*3
No of ways to answer 2 questions = ⁵C₂*3² (For each question there are 3 different choices to answer)
No of ways to answer 3 questions = ⁵C₃*3³
No of ways to answer 4 questions = ⁵C₄*3⁴
No of ways to answer 5 questions = ⁵C₅*3⁵
So, Total number of ways = ⁵C₁*3 + ⁵C₂*3² +  ⁵C₃*3³ +  ⁵C₄*3⁴ + ⁵C₅*3⁵
= (1+3)⁵ - 1(From binomial theorem)
=1023