Combinations of two

The diagram shows that every combination of two elements chosen from $6$ on the line $n = 6$ corresponds to exactly one dot in the triangular array above.  This triangular array above the dotted line in the diagram represents the triangle number $T_5$. Conversely the diagram also shows that every dot in the triangular array for $T_5$ corresponds to one and only one of the choices of pairs of elements in the line $n=6$ . Hence the number of combinations of two elements chosen from $6$ is equal to $1 + 2 + 3 + 4 + 5$.

 

This generalises directly to finding the number of distinct pairs of elements chosen from $n$ elements. Draw the triangular array for $T_n$, that is rows of dots for $n = 1, 2, 3, ... n$. Now in the same way as for $n=6$, every pair of elements chosen from the bottom row can be joined by lines in the diagram to meet in a single dot in the array for $T_{n-1}$. Conversely each dot in $T_{n-1}$ corresponds to one and only one pair chosen from $n$ elements on the bottom line. This shows that the number of ways of choosing pairs from $n$ objects, generally denoted by $^nC_2$, is $$T_{n-1} = 1 + 2 + ...+ (n-1).$$ Putting two $T_{n-1}$ triangular arrays  side by side gives $n(n-1)$ dots so $$^nC_2 = T_{n-1} = \frac{1}{2}n (n - 1).$$