A multiple choice exam contains 10 questions There are four

A multiple choice exam contains 10 questions. There are four possible answers for each question. (a) How many ways can a student answer the questions on the exam if every question is answered? (b) How many ways can a student answer the questions on the exam if the student can leave answers blank? (c) Assuming that there is only one correct answer to each question, and assuming that a student does not leave any questions blank, how many of the possibilities from part (a) of this question will earn the student a 70% or higher on the exam if all questions are weighted equally.

Solution

A)Number of ways a student can answer the question on the exam if every question is answered=4¹⁰

B)Number of ways a student can answer the question on the exam he can leave blank=5¹⁰

Because we can treat leaving the question as one more option so we have total 5 options to answer.

C) To get 70 % student needs to give 7 correct answers

so now number of way to choose to the 7 correct answers out of the 10 question=10C₇

now think about the remaining 3 question

number of ways to answer the remaing 3 question=4³

So total number of ways to get exact 70 % = 10C₇ *4³

TO get 80 % ,student need to give 8 correct answers

so now number of way to choose to the 8 correct answers out of the 10 question=10C₈

now think about the remaining 2 question

number of ways to answer the remaing 2 question=4²

So total number of ways to get exact 80 % = 10C₈ *4²

TO get 90 % ,student need to give 9 correct answers

so now number of way to choose to the 9 correct answers out of the 10 question=10C₉

now think about the remaining 1 question

number of ways to answer the remaing 1 question=4¹

So total number of ways to get exact 90 % = 10C₉ *4¹

TO get 100 % ,student need to give all answers correct

so now number of way to choose to the 10 correct answers out of the 10 question=10C₁₀

so now total number of ways to get at least 70 %= 10C₇ *4³ +10C₈ *4² +10C₉ *4¹ +10C₁₀

so the number possibillities to get 70 % or higher on exam =10C₇ *4³ +10C₈ *4² +10C₉ *4¹ +10C₁₀