If $a,$ $b,$ and $c$ are real numbers such that $a + b + c = 4$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 5,$ find the value of
\[\frac{a}{b} + \frac{b}{a} + \frac{a}{c} + \frac{c}{a} + \frac{b}{c} + \frac{c}{b}.\]
This problem looks like it involves some algebraic manipulation and possibly some clever use of symmetry or factoring. I notice that the expression I want to find is symmetric in $a,$ $b,$ and $c,$ meaning that swapping any two of them does not change the value. This makes me wonder if I can somehow use the fact that their sum and the sum of their reciprocals are both given. One thing I can try is to multiply the first equation by the second equation and see what happens. This gives me $(a + b + c)(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = 20,$ which I can expand using the distributive property. This gives me $1 + \frac{a}{b} + \frac{a}{c} + \frac{b}{a} + 1 + \frac{b}{c} + \frac{c}{a} + \frac{c}{b} + 1 = 20.$ I see that this has the expression I want to find, plus three extra ones. So, if I subtract three from both sides, I get $\frac{a}{b} + \frac{b}{a} + \frac{a}{c} + \frac{c}{a} + \frac{b}{c} + \frac{c}{b} = 17.$ 