With the digits listed from least to greatest, the $3$-digit integers are $138,146,226,234.$ $226$ can be arranged in $3$ ways, and the other three can be arranged in $6$ ways. There are $3+6(3)=(\mathbf{D}) 21$ 3-digit positive integers. 
How many $3$-digit positive integers have digits whose product equals $24$ ? (2009 AMC 8 Problem, Question \#16) 
Options:
(a) $$12$$ 
(b) $$15$$ 
(c) $$18$$ 
(d) $$21$$ 
(e) $$24$$ 