How many ways to choose $n$ distinct objects from a set of $m$ objects?

Question

So if a set contains $n$ objects, how many different ways would it be possible to pick distinct $m$ elements that are predetermined from a total of $c$ choices?

E. g. $\{1,2,3,4,5,.....,n\}$, how many ways would it be possible to pick $1$ and $2$ and $3$ (in this case $m=3$) if $c$ elements are picked (no duplicates)?

Answer 1

Lets say you have $n$ elements and some particular $m$ objects are fixed (they must be chosen).

There are a total of $c$ elements that need to be chosen with $m$ elements that must be there.

Now this means that there are $c-m$ elements that must be selected from $n-m$ elements.

$$\binom{n-m}{c-m}$$