# How do you find the coefficient of x in the expansion of (x+3)^5?

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Euan S. Share
Jul 19, 2016

Coefficient is 405.

#### Explanation:

We use the binomial theorem:

${\left(x + y\right)}^{n} = {\sum}_{k = 0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) {x}^{n - k} {y}^{k}$

where ((n),(k)) = (n!)/(k!(n-k)!)

For $n = 5$ and $y = 3$ we are looking for the coefficient of ${x}^{1}$. This means we need $n - k = 1 \implies k = 4$.

((5),(4))x^1*3^4 = (5!)/(4!1!)*81*x = 405x

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