In the binomial expansion of $(a + 2x)^{7}$ where $a$ is a constant the coefficient of $x^{4}$ is 15120 Find the value of $a.$ - Edexcel - A-Level Maths Pure - Question 6 - 2020 - Paper 2

The general term in the binomial expansion of $(a + 2x)^{7}$ is given by:

$T_k = \binom{n}{r} a^{n-r} (2x)^r$

where $n = 7$ and $r = 4$. So,

$T_4 = \binom{7}{4} a^{7-4} (2x)^4 = \binom{7}{4} a^{3} (2^4 x^4)$

Calculating \binom{7}{4}$, we have:

$\binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} = 35$

Thus, the term becomes:

$T_4 = 35 a^{3} (16 x^4) = 560 a^{3} x^4$