MHB Count the number of strings of length 8...

AI Thread Summary
The discussion revolves around counting the number of strings of length 8 formed from the set A = {w, x, y, z}, specifically those that start with either w or y and contain at least one x. The initial confusion stems from the interpretation of the question and the elements of set A. The correct approach involves calculating the total combinations starting with w or y and then subtracting those that do not include x. The final calculation arrives at a total of 28,394 valid strings. This problem highlights the importance of combinatorial reasoning in string formation.
shamieh
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Count the number of strings of length $8$ over $A = \{w, x, y, z\}$ that begins with either $w$ or $y$
and have at least one $x$


I don't understand this question at all. First of all, this is a set A that contains 4 elements $w,x,y,z$ correct? They are asking me to count the number of strings of length 8? None of these are length 8, what are they asking me? Also wouldn't this be:

$A = \{ wxyz, wzxy, wyxz, ... \}$ I mean I'm going to have a lot of different combinations right? Also, every single group contained in the set is going to have at least one $x$.. I am so confused here.
 
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Wait.. am I making this harder than it is? Is that just $4^8$
 
|Ok I think I've figured it out.. Can someone check my work? Sorry to be a pest with the triple posts...

$A = \{w,x,y,z\}$

$U = \{w,y\} * A^7$

$S = \{w,y\} * \{w,y,z\}^7$

$|U - S| = |U| - |S|$

$= |\{w,y\} * A^7| - |\{w,y\} * \{w,y,z\}^7|$

$= |\{w,y\}||A|^7 - |\{w,y\}||\{w,y,z\}|^7$

$= 2(4^7) - 2(3^7) = 28,394$
 
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