Are You Sure We Can Do This? (Part 2)

Here we have to use the Extended Binomial Theorem which is valid for all real numbers.

It says that :

$\dbinom{n}{r}$ = $\frac{n(n-1)(n-2).......(n-r+1)}{r!}$

We put n as negative to get :

$\dbinom{-n}{r}$ = $\frac{-n(-n-1)(-n-2).......(-n-r+1)}{r!}$

This can be written as :

$\dbinom{-n}{r}$ = $\frac{(-1)^r(n(n+1)(n+2).......(n+r-1))}{r!}$

By simplifying we get:

$\dbinom{-n}{r}$ = $\frac{(-1)^r(n+r-1)!}{r!(n-1)!}$

Therefore this can be written as :

$\dbinom{-n}{r}$ = $\dbinom{n+r-1}{r}(-1)^r$

Now we can use this formula for solving the question(n=17 , r=9).

$\dbinom{-n}{r}$ = $\dbinom{17+9-1}{9}(-1)^9$

= $\dbinom{25}{9}(-1)$

= $\boxed{-2042975}$