The number of distinct terms in a trinomial expansion

Question

$$(x^3 + \frac1{x^3} +1)^{200}$$ is the given expression. How many distinct terms are in this expression when expanded. I know that there are a total of $3^{200}$ terms before combining the terms but I'm struggling to find the distinct terms. Please provide a clear explanation of the answer. Thanks in advance.

Answer 1

The answer should be $401$: in the complete expansion of the polynomial above, there are no negative coefficients, there will be no cancellation, so you just need to count how many powers of $x$ appear. Substituting $x^3$ with $y$, it is clear that every power of $y$ between $y^{200}$ and $y^{-200}$ appears. Hence, the answer is $401$.

Answer 2

I'm going to simplify the problem a bit to explain my method, then you can apply it to this one:

Suppose instead we had:

$$(x+1)^{200}$$

This is $(x+1)(x+1)(x+1)...$

Suppose we wish to find any term (from the $2^{200}$ in the expanded equation, before collecting like terms). We could do this by going through each of the brackets one at a time and selecting either $x$ or $1$. Once we go through every bracket selecting one of the terms, the product of the terms we picked out is one of the terms of the expansion.

Let's suppose we pick $x$ a total of $t$ times, and therefore we pick $1$ a total of $(200-t)$ times. Then the power, $p$, of a given term is:

$$p=(1)(t)+0(200-t)=t$$

Applying this to your equation, can you see we have:

$$p=(3)(t)+(-3)(u)+(0)=3(t-u)$$ This makes it obvious that the powers are only multiples of $3$, so you just need to figure out how many are in your range.

Answer 3

Does this work?

You can take a product of $ k_1 $ terms in $ x^3 $, $ k_2 $ terms in $ x^{-3} $ and $ k_3 $ terms in $ x^0 $, where the same power of $ x $ can be obtained in multiple ways. In doing this we require $ k_1+k_2 +k_3 = 200 $ and $ 0 \leq k_1, k_2, k_3 \leq 200 $. Consolidating terms of the same power of $ x $ then results in gathering terms of the form $ x^{3(k_1-k_2)} $. This yields all distinct powers of $ x^3 $ from $ -200 $ to $+200 $, including zero, implying 401 distinct terms in the expansion. We note that none cancels since the individual coefficients are all 1.