1. Expansion

What does the expansion of (x + y + z)^n look like when n = 1, 2, 3? and also how does this relate to the question: Find the number of strings of 3 nonnegative integers whose sum is 13 which I cannot figure out.

2. Go to Wikipedia and search for "multi-nomial theorem".

3. Each term of $\displaystyle \left( {x + y + z} \right)^n$ look like $\displaystyle \frac{{n!}}{{a!b!c!}}x^a y^b z^c \,,\,\left( {a + b + c = n} \right)$.

As to your question “Find the number of strings of 3 nonnegative integers whose sum is 13 which I cannot figure out.” I hope that you have not confused two different ideas. The answer to that question is the value of the coefficient of $\displaystyle x^{13}$ in the expansion of $\displaystyle \left( {\sum\limits_{k = 0}^{13} {x^k } } \right)^3$.

The "Trinomial Expansion" . . . an interesting concept.

What does the expansion of $\displaystyle (x + y + z)^n$ look like when $\displaystyle n = 1, 2, 3$?

When I was in college, I expanded some trinomials and made some "discoveries".
It's best to write the terms in a triangle.

$\displaystyle (x + y + z)^1 \;=\;\begin{array}{ccc}& x & \\ \\ y & & z \end{array}$

$\displaystyle (x+y+z)^2 \;=\;\begin{array}{ccccc}& & x^2 & & \\ \\ & 2xy & & 2xz & \\ \\ y^2 & & 2yz & & z^2 \end{array}$

$\displaystyle (x+y+z)^3 \;=\;\begin{array}{ccccccc}& & & x^3 \\ \\ & & 3x^2y & & 3x^2z \\ \\ & 3xy^2 & & 6xyz & & 3xz^2 \\ \\ y^3 & & 3y^2z & & 3yz^2 & & z^3\end{array}$

$\displaystyle (x+y+z)^4 \;=\;\begin{array}{ccccccccc}& & & & x^4 \\ \\ & & & 4x^3y & & 4x^3z \\ \\ & & 6x^2y^2 & & 12x^2yz & & 6x^2z^2 \\ \\ & 4xy^3 & & 12xy^2z & & 12xyz^2 & & 4xz^3 \\ \\ y^4 & & 4y^3z & & 6y^2z^2 & & 4yz^3 & & z^4 \end{array}$

There are some remarkable patterns.

Reading down the left edge, we have: .$\displaystyle (x + y)^n$
Reading down the right edge, we have: .$\displaystyle (x+z)^n$
Reading across the bottom, we have: .$\displaystyle (y+z)^n$

For $\displaystyle (x+y+z)^n$, the coefficient of $\displaystyle x^ay^bz^c$ is, of course: .$\displaystyle {n\choose a,b,c}$