Question 1207776
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How many random ways can a 20-question multiple-choice test be answered, 
with each question having 4 options?
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<pre>
Imagine that the option are A, B, C, D  (i.e., the options are marked by letter A, B, C, and D).


Then one whole answer to 20 multiple choice question is a 20-letters word, written in the alphabet A, B, C, D.


Different answers create different words, and different words correspond to different answers.


So, the problem's question is the same as to ask: how many 20-letters words 
do exist in 4-letter alphabet A, B, C, D.


The letters in different positions are independent; in each position, any of 4 letters can be.


So, the <U>ANSWER</U> is:  {{{4^20}}} = 1099511627776  different 20-letter words are possible 
in the alphabet A, B, C and D.


Hence,  {{{4^20}}} = 1099511627776  different answers are possible in this multiple choice test.
</pre>

Solved.


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This model using words of the certain length written in some alphabet is useful for solving such problems 
on multiple choice tests, and for solving many other combinatorics problems, where there is 
a finite number of independent trials with the fixed number of independent outcomes for each trial.



To see many other similar &nbsp;(and different) &nbsp;problems of this kind, &nbsp;look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Combinatoric-problems-for-entities-other-than-permutations-and-combinations.lesson>Combinatoric problems for entities other than permutations and combinations</A> 

in this site.