Isn't Combination just selection?

Question

So, There is this question which I came across. "Total no of handshakes among 15 people." The answer seems to be just ₁₅C₂. Isn't that just a way of selecting and not the number of handshakes? I'm in grade 10th. So don't judge.

Answer 1

I mean, $_{15}C_2$ gives a numerical result as well even if describes a method of selection:

$$_{15}C_2 = \frac{15!}{2! \cdot (15-2)!}$$

and generally

$$_{n}C_r = \frac{n!}{r! \cdot (n-r)!}$$

Do the calculations and boom, a number. $_{n}C_r$ is just a notation representing the underlying number (or, rather, the calculations leading up to that number), because who wants to write that fraction every single time?

Answer 2

$_{15}C_2$ is the number of ways to select 2 people from 15 people (when order doesn't matter). This is exactly the number of handshakes that occur if $15$ people all shake hands with each other. There is one handshake for each set of 2 people.