Question

What is the coefficient of x^4 in expansion of (2x+1)^4 ?

Answer 1

The expansion would be 16x^4+32x^3+24x^2+8x+1
The coefficient would be 16

Hope I didn't mess up for your sake

Answer 2

Consider the typical binomial expansion: (1 + x)ⁿ

$(1 + x)^{n} = \sum_{r=0}^{n} \left(\begin{array}{ccc}n\\r\end{array}\right)(1)^{n - r}(x)^{r}$

$\therefore (1 + 2x)^{4} = \sum_{r=0}^{4} \left(\begin{array}{ccc}4\\r\end{array}\right)(1)^{4 - r}(2x)^{r}$

Since we need to find the coefficient of x⁴, then we need to let r = 4 in order to find the coefficient.

Thus, the coefficient is:
$\left(\begin{array}{ccc}4\\4\end{array}\right)(1)^{4 - 4}(2x)^{4}$
$= 16x^{4}$

Thus, the coefficient of x⁴ is 16.