0
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

Here is a start to the problem:

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.