For some real number $a$ and some positive integer $n$, the first few terms in the expansion of $(1 + ax)^n$ are \[1 - 20x + 150x^2 + cx^3 + \dotsb.\]Find $c$.
The expansion of $(1 + ax)^n$ can be written using the binomial theorem as:
$$(1 + ax)^n = \sum_{k=0}^{n} \binom{n}{k} (ax)^k$$
The first few terms in this expansion are:
$$(1 + ax)^n = \binom{n}{0} + \binom{n}{1}(ax) + \binom{n}{2}(ax)^2 + \binom{n}{3}(ax)^3 + \cdots$$
Simplifying the first three terms:
$$(1 + ax)^n = 1 + nax + \frac{n(n-1)}{2}(ax)^2 + \cdots$$
We are given that the first three terms are:
$$1 + 20x + 150x^2 + cx^3 + \cdots$$
Comparing the coefficients of $x^3$ on both sides of the equation, we get:
$$\binom{n}{3}(a)^3 = c$$
To find $c$, we need to find $\binom{n}{3}$ and $a$. We can use the first two terms of the expansion to solve for $a$ and $n$.
From the given first two terms, we have:
$$1 + nax = 1 + 20x$$
This implies that $nax = 20x$ or $a = \frac{20}{n}$.
Substituting this value of $a$ in the expression for the second term:
$$20x = \binom{n}{1} \left(\frac{20}{n}\right)x$$
Simplifying this equation gives:
$$n = 2\binom{n}{1} = 2n$$
Thus, $n = 2$.
Now, we can use $n=2$ and $a=\frac{20}{n}=10$ to find $c$:
$$\binom{2}{3}(10)^3 = 2000$$
Therefore, $c=2000$.
