Question

# How many strings of eight English letters are there a) that contain no vowels, if letters can be repeated? b) that contain no vowels, if letters cannot be repeated? c) that start with a vowel, if letters can be repeated? d) that start with a vowel, if letters cannot be repeated? e) that contain at least one vowel, if letters can be repeated? f) that contain exactly one vowel, if letters can be repeated? g) that start with X and contain at least one vowel, if letters can be repeated? h) that start and end with X and contain at least one vowel, if letters can be repeated?

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DEFINITIONS

$\textbf{Product rule }$If one event can occur in $m$ ways AND a second event can occur in $n$ ways, the number of ways the two events can occur in sequence is then $m\cdot n$.

$\textbf{Subtraction rule }$ If an event can occur either in $m$ ways OR in $n$ ways (overlapping), the number of ways the event can occur is then $m+n$ decreased by the number of ways that the event can occur commonly to the two different ways.

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