Answer to Question #152481 in Statistics and Probability for Yosef

Question

A QCM contains 10 questions, the student has to choose only one correct answer from 3 possible answers. How many different responses can we get?

Answer 1

Let u_n be the number of all possible responses we can get, if QCM contains n questions.

Let's show then, that for all n u_n+1 = 3u_n.

Indeed, any response for n+1 questions consists of two parts: a response to the first n questions (total of u_n variants) and an answer to the last one (3 variants). All variants from the first part are compatible with any variant from the second part. Therefore, the total number of responces u_n+1 is a product: u_n * 3.

The sequence u₁, u₂, ..., u_n is a geometric progression with the common ratio 3 and, by the formula of the n-th term of it, we have u_n = u₁*3^n-1. So, u₁₀ = 3*3⁹ = 3¹⁰ = 59049.

Answer: There exist 59049 different responses.