A multiple choice test has 15 questions with 1 correct answer and 3 incorrect answers for each.

Question

How many ways can the test be answered if

 

a) all questions are answered

b) exactly two questions are left blank?

c) exactly three questions are answered incorrectly?

d) exactly 10 questions are answered incorrectly?

 

I have the answers but honestly have no idea how my friend got them. I understand part a because there are 4 answer choices and 15 questions so you have 4¹⁵ possible ways but im lost for part b-d.

 

I have a test tomorrow over this stuff :(

Answer 1

this type of question seems motivated by how to tell if students are cheating, as one student may copy another student's false answers. if odds of the particular answers are small enough, it's strong evidence of cheating

a) all 15 answered: 4^15= 1.07374182 x 10^9

x

b) exactly 2 left blank: 15C2 = 15!/13!2! = 15x14/2 = 15x7 = 105 ways to leave exactly 2 blank. multiply that times 4^13 = 105 x 67,108,864= 7.04643072 x 10^9

c) exactly 3 answered incorrectly: 3 ways to get a question wrong. 3x3=9 x 12 = 108

d) exactly 10 answered incorrectly: 3x10 = 30 x 2 = 60