How can I find the number of dissimilar or distinct terms in $(1+x + x^2 +x^3)^n$?

Question

How can I find the number of dissimilar or distinct terms in $(1+x + x^2 +x^3)^n$ ?

I know it would be $\binom{n+r-1}{r-1}$ when they are $a , b, c ,d$ instead of $1 , x , x^2 , x^3$.

But how can I do it for this case? Can anyone please give me a hint?

Answer 1

What's the highest degree monomial you can get? What's the lowest degree? Can you get the ones in between?

Answer 2

This isn't a very good answer, but I suppose you asked for a hint.

When you say distinct terms I assume you are referring to the order of x. so:

$x^2 \ \ and \ \ 2x^2$

are not distinct, but:

$x^2 \ \ and \ \ 2x^3$

are distinct.

So you want to know the number of distinct terms in a polynomial. Unfortunately this polynomial is difficult to expand as we aren't sure what n is. Hopefully we can get an answer in terms of n.

Here is how I would attempt the problem: differentiate your polynomial, which means you lose your constant term (in this case its 1 regardless of what n is), so now you have 1 less distinct term than in your original polynomial. Now differentiate again, you have now lost your original linear terms, so you have 2 less distinct terms than in your original polynomial.

So maybe it is possible to figure out how many times do you need to differentiate this polynomial (as a function of n) before it becomes zero. I think that number will be how many distinct terms you have.

You will need the product rule and the chain rule, and will probably require some summation notation. Good luck!