<s>[INST] Find the coefficient of $x^{30}$. Find the coefficient of $x^{30}$ in the given polynomial
$$
\left(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}\right)^5
$$
I don't know how to solve problems with such high degree.
 [/INST] The coefficient attached to $x^{30}$ will be the number of ways you can add up to $30$ by using the numbers $0$-$12$ up to five times. (Here order matters)
For instance $1+1+2+10+6=30$ is one way.
$10+10+10+0+0=30$ is another and so is $0+10+10+10+0 = 30$.
The reason for this is more apparent for smaller polynomials.
For instance let's calculate the coefficient of $x^4$ in $(1+x+x^2)^3$.
$$(1+x+x^2)(1+x+x^2)(1+x+x^2)$$
Eventually we will have every combination of products between the three terms. We can achieve $x^4$ is several ways:
$$1\cdot x^2 \cdot x^2$$
$$x^2 \cdot 1 \cdot x^2$$
$$x^2 \cdot x^2 \cdot 1$$
$$x \cdot x \cdot x^2$$
$$x \cdot x^2 \cdot x$$
$$x^2 \cdot x \cdot x$$
Thus we have six ways of achieving $x^4$, and the powers of $x$ in each of these cases add up to $4$. Therefore the coefficient of $x^4$ is six.
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