Question

If a multiple-choice test consists of 5 questions, each with 4 possible answers of which only 1 is correct, (a) in how many different ways can a student check off one answer to each question?

(b) in how many ways can a student check off one answer to each question and get all the answers wrong?

Answer 1

(a) 1024 ways

(b) 243 ways

Explanation:

When answers are independent, the total number of ways all questions can be answered is the product of the number of ways each question can be answered.

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(a)

The first question can have one answer 4 ways. The second question can be answered 4 ways for each of the ways the first question was answered. That is, the two questions can be answered 4×4 = 16 ways. Similarly, each additional question multiplies the number of ways the questions can be answered by 4.

The 5 questions can be answered 4×4×4×4×4 = 4^5 = 1024 ways.

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(b)

There are 3 ways to answer the first question incorrectly. Each additional question can be answered incorrectly 3 ways for each of the ways previous questions were answered. The total number of ways all questions can be answered incorrectly is ...

3^5 = 243

A student can answer all questions wrong in 243 different ways.