Question 1173626: A five item multiple choice quiz offers 4 options (A,B,C,D) for each item. How many ways can this quiz be answered such that at least 4 items are correct.
Found 2 solutions by ikleyn, ewatrrr:
Answer by ikleyn(52780)   (Show Source):
You can put this solution on YOUR website! .
A five item multiple choice quiz offers 4 options (A,B,C,D) for each item.
How many ways can this quiz be answered such that at least 4 items are correct.
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"at least 4 items are correct" means that in four of these 5 items, the answer is PECISELY as NEEDED, in correct position.
So, there are  = 5 ways to select these correctly answered items;
and then for the single remaining item you have 4 options to answer incorrectly.
It gives 5*4 = 20 ways to have exactly 4 items answered correctly.
Add to it the case when ALL FIVE items are answered correctly, and you will get the ANSWER to the problem's question
20 + 1 = 21.
Solved.
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The post-solution note:
@ewatrrr produced different answer from mine, and it REQUIRES me to make explanations.
The cause is in that @ewatrr INCORRECTLY treats the notion "answer".
In this context, the answer IS NOT HOW or WHAT the teacher evaluates the student's answer.
It is not a "CORRECT" or "INCORRECT" output value of the item.
For each "item", the answer is one of options A, B, C, or D, marked by student and treated as the answer.
It is absolutely standard and traditional interpretation in this kind of problems.
Keeping it in mind, EVERYBODY should CLEARLY UNDERSTAND that my solution is CORRECT,
while @ewatrrr solution is WRONG.
They are VERY ELEMENTARY conceptions for this kind of problems, that everybody knows or should know --- but not @ewatrrr.
Answer by ewatrrr(24785)  (Show Source):
You can put this solution on YOUR website!
Hi,
A five item multiple choice quiz offers 4 options (A,B,C,D) for each item.
How many ways can this quiz be answered such that at least 4 items are correct.
With 4 Correct Answers 0r 5 Correct Answers
Wish You the Best in your Studies.
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