Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$

It's straightforward to show that

$$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$

but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first $n$ natural numbers count the number of ways I can choose a pair out of $n+1$ objects? What's the intuition behind this?

Consider a tournament with $n+1$ teams each playing each other. We will count the number of matches played in two ways.

• Every match is played between two teams. This inturn implies that the number of matches is $\dbinom{n+1}2$.
• We will now count the number of distinct matches played team by team.
  • The number of matches played by the first team is $n$.
  • The number of matches played by the second team is $n-1$, since their match with the first team has already been accounted for.
  • The number of matches played by the third team is $n-2$, since their matches with the first and second team have already been accounted for.
  • The number of matches played by the $k^{th}$ team is $n-k+1$, since their matches with the first $k-1$ teams have already been accounted for. Hence, the total number of matches is $$n+(n-1) + (n-2) + \cdots + 1$$

Suppose that you want to choose a subset $\{m,n\}$ with two elements of the set $$ \{1,2,\dotsc,n+1\} $$ Count this in two ways one of them naturally equals $\binom {n+1}2$ and for the other observe that

If $max\{m,n\}=2$ then we have one subsets $\{m,n\}$.
If $max\{m,n\}=3$ then we have two subsets $\{m,n\}$.
$\vdots$
If $max\{m,n\}=n+1$ then we have $n$ subsets $\{m,n\}$.

Now add up these cases to derive the identity.$\square$

This is the classic proof without words, from https://maybemath.wordpress.com/

That doesn't help with this part of your question:

Why does the sum of the first $n$ natural numbers count the number of ways I can choose a pair out of $n+1$ objects?

Here's a way to rephrase @user17762 's excellent accepted answer.

Imagine $n+1$ kids in a room. Each shakes hands with all the others. Then each kid shakes hands $n$ times, so there are $n(n+1)$ handshakes - each counted twice. You can pick a pair of kids (that is, a handshake) in $n(n+1)/2$ ways. But you can also think about the kids shaking hands as they enter the room one at a time. The second kid coming has one hand to shake. The third has two, and so on, for a total of $1 + 2 + \cdots + n$.

The intuition is that for the pairs can be listed in the following way.

$$\begin{array}{ccccccc} 1,2 & & & & & & \\ 1,3 & 2,3 & & & & & \\ 1,4 & 2,4 & 3,4 & & & & \\ 1,5 & 2,5 & 3,5 & 4,5 & & & \\ 1,6 & 2,6 & 3,6 & 4,6 & 5,6 & & \\ 1,\vdots & 2,\vdots & 3,\vdots & 4,\vdots & 5,\vdots &\ddots & \\ 1,n+1 & 2,n+1 & 3,n+1 & 4,n+1 & 5,n+1 & \cdots & n,n+1 \\ \end{array}$$

Notice that each row has length $i$ for $i=1,\ldots,n$ since the number of pairs with maximum element $i+1$ is $i$. Therefore the total number of pairs, which is $\binom{n+1}{2}$ is $\displaystyle \sum_{i=1}^n i$.

If you want to choose a pair out of $n+1$ objects (for example, $\{0,1,\dots,n\}$), the possibilities are:

$\{0,1\}$, $\{0,2\}$, ..., $\{0,n\}$, giving $n$ possibilities.

$\{1,2\}$, $\{1,3\}$, ..., $\{1,n\}$ giving $n-1$ possibilities. (note that we've already picked $\{1,0\}$, so we can't repeat it here)

$\{2,3\}$, $\{2,4\}$, ..., $\{2,n\}$ giving $n-2$ possibilities.

$\ \ \ \ \vdots$

$\{n-2,n-1\}$, $\{n-2,n\}$ giving $2$ possibilities.

$\{n-1,n\}$ giving $1$ possibility.

So the number of pairs is $n+(n-1)+\dots+2+1$