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If you were given a five-question multiple choice test that has four possible answers for each question (a,b,c,d),
how many ways could you answer all the questions on the test if you do not put down the same answer for two consecutive questions?
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1. The total space T of all possible answers (including all allowed and all non-allowed) is the set of all 5-symbol words {x,y,z,u,w},
where each symbol x, y, z, u, and w can be any of 4 letters a, b, c, or d.
The cardinality (the number of elements) of this set is .
2. The sub-space S of non-allowed answers are those 5-symbol words {x,y,z,u,w} that have coinciding symbols in two CONSECUTIVE positions.
You can imagine this sub-space as the union of 5-symbol words of the form
S = {x,x,y,z,w} U {y,x,x,z,w} U {y,z,x,x,w} U {y,z,w,x,x}
By gluing two coinciding symbols in one, you can see that the set S is the same (is isomorphic) to the set of all 4-symbol words {x,y,z,w},
where each symbol x, y z and w can take any of 4 values a, b, c and d.
From it, it is clear that the cardinality (the number of elements) of the sub-set S is .
3. Now it is clear that the allowed answers represent the set T \ S, and its cardinality is - = 768.
Answer. There are - = 768 allowed answers.