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

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?

Answer

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
    $$n+(n-1) + (n-2) + \cdots + 1$$