Counting length-8 strings from {w,x,y,z} starting with w or y and containing x

[shamieh]

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.

 

[shamieh]

Wait.. am I making this harder than it is? Is that just $4^8$

 

[shamieh]

|Ok I think I've figured it out.. Can someone check my work? Sorry to be a pest with the triple posts...

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

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

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

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

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

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

$= 2(4^7) - 2(3^7) = 28,394$