Need some explanation on the combinatorics formula $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$

Question

$\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$ is a famous recursive equation for computing $\binom{n}{r}$, which means number of ways for selecting $r$ objects from $n$ objects.

I have read in high school the logical meaning of this formula.

I remember my teacher saying that in order to calculate number of ways of selecting $r$ objects from $n$ objects, we can either discard object at $r$ position using $\binom{n-1}{r-1}$, or we can include the object at $r$ position using $\binom{n-1}{r}$.

But I don't understand this logic now.

Can someone please explain me this with some example or any editorial?

Answer 1

Let's say you have $n$ distinct objects $A=\{1,2,...,n\}$. Now, if you choose a set $S\subset A$ of $r$ elements, it either contains the element $1$ or not. If it has $1$, then you need to choose the remaining $r-1$ elements from the set $\{2,3,...,n\}$,which is ${n-1}\choose{r-1}$. If it does not contain $1$, then you have to choose all the $r$ elements from $\{2,...,n\}$, which is ${n-1}\choose{r}$. Finally, there would be no intersection between cases, so, they should add up to give ${n}\choose{r}$