Distinguishable telephone poles being painted Each of n (distinguishable) telephone poles is painted red, white, blue or yellow.  An odd number are painted blue and an even number yellow.  In how many ways can this be done?
Can some give me a hint how to approach this problem?

The exponential generating function (EGF) for the number of red or white poles is
$$1 + z + \frac{1}{2!}z^2 + \frac{1}{3!}z^3 + \dots = e^z$$
The EGF for the number of blue poles is
$$z + \frac{1}{3!}z^3 + \frac{1}{5!}z^5 + \dots = \frac{1}{2}(e^z-e^{-z})$$
The EGF for the number of yellow poles is
$$1 + \frac{1}{2!}z^2 + \frac{1}{4!}z^4 + \dots = \frac{1}{2}(e^z + e^{-z})$$
So the EGF for the number of acceptable permutations of red, white, blue and yellow poles is
$$f(z) = (e^z)^2 \cdot \frac{1}{2}(e^z-e^{-z}) \cdot \frac{1}{2}(e^z + e^{-z}) = \frac{1}{4}(e^{4z}-1)$$
The number of acceptable permutations of $n$ poles is the coefficient of 
$\frac{1}{n!}z^n$
in $f(z)$, which by inspection is $4^{n-1}$ for $n > 0$.
