Find the coefficient of $x^{10}$ in $(x + (\frac{1}{x}))^{100}$

Question

Find the coefficient of $x^{10}$ in $(x + (\frac{1}{x}))^{100}$

---

My solution:

We can calculate the coefficient using the Binomial Theorem:

$$(x+y)^n = \sum_{k=0}^n\binom{n}{k}x^{n-k}y^k$$

We know that:

• $n = 100$
• $k = 90$, since $100-90 = 10$ and we want $x^{10}$

thus $$\binom{100}{90}x^{100-90}+(\frac{1}{x})^{90} = \binom{100}{90}x^{10}(\frac{1}{x})^{90}$$

So the coefficient of $x^{10}$ is $\binom{100}{90}$

Is this correct?

Answer 1

We have

$$(a+b)^n=\sum_{k=0}^n{n\choose k} a^{n-k}b^k$$ so with $a=x$ and $b=\frac1x$ we get the coefficient of $x^{10}$ on $100-2k=10$ so $k=45$ and then the coefficient is ${100\choose 45}$.