This question is from a the textbook Mathematical Statistics by John Rice. I found a solution online for this question but am having a difficult time understanding the steps they took.
In the line beginning with $P(X=x)$, the next step after, they switch the starting index from $0$ to $x$, what was the intuition or idea behind doing something like this or was this just an A-ha! moment? If I was in a test environment, I would've never been able to think of something like this much less be able to change the contents of the summation to match the one above.
Also, is there an alternative, ideally simpler, solution to this question?
The line in the text that includes the multiplicative term $n\choose x$ is defined to be zero whenever $x>n$. The definition can be written as follows:
$${n\choose x}=\frac{n (n-1) (n-2)\cdots (n-x+1)}{x!}$$
So
$${3\choose 4}=\frac{3\times (3-1)\times (3-2)\times (3-3)}{4!}=\frac{3\times 2\times 1\times 0}{4!}=0$$
The line in the text could have started out with $n=x$ as the summation index but chose to let the definition of a binomial coefficient drop out the appropriate terms.