Question

# How many positive integers between 100 and 999 inclusivea) are divisible by 7?b) are odd?c) have the same three decimal digits?d) are not divisible by 4?e) are divisible by 3 or 4?f) are not divisible by either 3 or 4?g) are divisible by 3 but not by 4?h) are divisible by 3 and 4?

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DEFINITIONS

$\textbf{Division rule }$If a finite set $A$ is the union of pairwise disjoint subsets with $d$ elements each, then $n=\dfrac{|A|}{d}$

$\textbf{Product rule }$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.

$\textbf{Sum rule }$ If an event can occur either in $m$ ways OR in $n$ ways (non-overlapping), the number of ways the event can occur is then $m+n$.

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