Let’s first write relevant definitions for this exercise:
THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are n1n2 ways to do the procedure.”
THE SUM RULE: “If a task can be done either in one of n1 ways or in one of n2 ways, where none of the set of n1 ways is the same as any of the set of n2 ways, then there are n1 +n2 ways to do the task.”
The definitions were taken from the textbook Discrete Mathematics and its Applications by Rosen.
Now we can solve the exercise.
a) In how many ways can a student answer the questions on the test if the student answers every question?
There are four possible answers to each question and the student will answer all the questions.
The student can answer the first question in 4 ways. After that, he/she can answer the second question in four ways, and so on. Notice that for each of the four possible answers to the first question, there are four possible answers to the second one, and so on.
So, we apply the product rule.
4x4x4…x4 (10 times, as there are 10 questions) = 410
Answer:
By the product rule, a student can answer the questions on the test in 410 ways.
b) In how many ways can a student answer the questions on the test if the student can leave answers blank?
The reasoning in this exercise is similar to the previous one.
In this case, we just need to notice that leaving an answer in blank, is another possible way to “answer it”.
So, in this case, there are five possible ways of answering each question. In other words, the student can choose any of the four possible answers, or leave it blank, which makes five possible answers.
Answer:
By the product rule, a student can answer the questions on the test in 510 ways.
Related exercises:
- How many bit strings are there of length eight?
- How many different three-letter initials with none of the letters repeated can people have?
- There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?
- A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?
- How many different three-letter initials are there that begin with an A?
