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.
~~~~~~~~~~~~~



<pre>
"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 {{{C[5]^4}}} = 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 <U>ANSWER</U> to the problem's question


      20 + 1 = 21.
</pre>

Solved.



/\/\/\/\/\/\/\/


<H3>The post-solution note:</H3>


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@ewatrrr produced different answer from mine, &nbsp;and it  &nbsp;<U>REQUIRES &nbsp;me</U>  &nbsp;to make explanations.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The cause is in that @ewatrr &nbsp;INCORRECTLY &nbsp;treats the notion &nbsp;"answer".




&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In this context, &nbsp;the &nbsp;<U>answer</U> &nbsp;IS &nbsp;NOT &nbsp;HOW &nbsp;or &nbsp;WHAT &nbsp;the teacher evaluates the student's &nbsp;answer.

    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is not a  &nbsp;"CORRECT" &nbsp;or &nbsp;"INCORRECT" &nbsp;output value of the item.


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;For each &nbsp;"item", &nbsp;the &nbsp;<U>answer</U> &nbsp;is one of options &nbsp;A, &nbsp;B, &nbsp;C, &nbsp;or &nbsp;D, &nbsp;marked by student and treated as the answer.


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is absolutely standard and traditional interpretation in this kind of problems.




&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Keeping it in mind, &nbsp;EVERYBODY &nbsp;should &nbsp;CLEARLY &nbsp;UNDERSTAND &nbsp;that my solution is CORRECT,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;while @ewatrrr solution is &nbsp;WRONG.



They are &nbsp;VERY &nbsp;ELEMENTARY &nbsp;conceptions for this kind of problems, &nbsp;that everybody knows or should know --- but not &nbsp;@ewatrrr.