Combinatorics problem using generating exp functions 2

Question

Calculate the number of sequences of length n that are made of $1, 2, 3, 4$ so that the digits $1,2$ shows an even number of times, And the digit $3$ shows at least 1 time.

I've been given a clue to use generating exponential function method.

But I don't understand how to start, any hints or general approach would be great!

Thanks in advance!

Answer 1

Here is a start to the problem:

Let $\displaystyle g_e(x)=\underbrace{\big(1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots\big)^2}_{{a_1,a_2}}\underbrace{\big(x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\big)}_{{a_3}}\underbrace{\big(1+x+\frac{x^2}{2!}+\cdots\big)}_{{a_4}}$

$\displaystyle\hspace{.65 in}=\bigg(\frac{e^x+e^{-x}}{2}\bigg)^2\big(e^x-1\big)(e^x)$