How many subsets are there with exactly 8 elements in a set of 16 elements? Below is a problem I did which I believe I did correctly. I would like somebody to confirm that I did, or tell me where I went wrong.
Problem:
Consider a set with $16$ elements in it. How many subsets does it have with exactly $8$ elements?
Answer:
Let $c$ be the number of subsets with exactly $8$ elements.
\begin{align*}
c &= \frac{16(15)(14)(13)(12)(11)(10)(9)}{8!} \\
c &= \frac{2(15)(14)(13)(12)(11)(10)(9)}{7!} \\
c &= \frac{2(15)(2)(13)(12)(11)(10)(9)}{6!} \\
c &= \frac{2(15)(2)(13)(2)(11)(10)(9)}{5!} \\
c &= \frac{2(15)(2)(13)(2)(11)(10)(9)}{5(4)(3)(2)} \\
c &= \frac{2(3)(2)(13)(2)(11)(10)(9)}{4(3)(2)} \\
c &= 13(11)(10)(9) \\
c &= 12870
\end{align*}

Your work appears correct.
The answer is "$16$ choose $8$":
$$\binom {16}8=\dfrac{16!}{8!8!}$$
