<PRE><B>Question 1115937</B>
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A test has 6 multiple choice questions each having 4 possible answers. In how many ways can a student respond to the questions if:


&lt;pre&gt;
a)  All questions must be answered?    &lt;&lt;&lt;---=== &lt;U&gt;Answer&lt;/U&gt;.  {{{4^6}}} ways.


       &lt;U&gt;Solution&lt;/U&gt;


    For simplicity, let us assume that you write one of the 4 letters A, B, C or D as the possible answer to each question.

    Then you have 4 opportunities for each of 6 question, which gives the above answer.


    &lt;U&gt;Memorize&lt;/U&gt;: the space of events in this case is the space of all 6-letter words written in 4-letter alphabet.




b)  Not all questions must be answered?       &lt;&lt;&lt;---=== &lt;U&gt;Answer&lt;/U&gt;.  {{{5^6}}}.


       &lt;U&gt;Solution&lt;/U&gt;


    We can very easily reduce this problem to the previous scheme by making an agreement that we write the &lt;U&gt;fifth&lt;/U&gt; symbol/letter  &quot;E&quot;  

    where the question is unanswered.


    Then the space of events is the space of all 6-letter words written in 5-letter alphabet, which gives the above answer.
&lt;/pre&gt;

Solved.


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The space (the set) of  n-symbol words over the m-symbol alphabet is the third entity widely used in Combinatorics after Permutations and Combinations.



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