How to determine the number of triads from a set of pairs?

Question

I have a set of paired numbers as follows:

 { 
   {{1,2},{1,3},{1,4},{1,5}},  
   {{2,3},{2,4},{2,5}},   
   {{3,4},{3,5}},  
   {4,5}  
 }

• (a) How can I calculate the number of unique triads, such that only 3 digits appear in each triad? In the example above, one triad would be: {{1,2},{1,3},{2,3}}. So in the set above, we would have 10 such sets.

• (b) What is the general formula, given $n$ number of items (or numbers, if you like) for the number of triads?

• (c) What is the correct mathematical language to describe this problem?

The number of unique pairs is given by the shifted triangular series: $k = n(n-1)/2$ = 10.
It can also be obtained through the ${n \choose m} = {5 \choose 2} = 10$

---

Possibly related questions:

• How many triangles from a set of pairs?

---

Clarification:

The answer for the 10 triads I am looking to calculate are:

{ 
  {12,13,23}, {12,14,24}, {12,15,25}, {13,14,34}, {13,15,35}, {14,15,45},  
  {23,24,34}, {23,25,35}, 
  {34,35,45}
}

Answer 1

Thanks to André Porto's comment, I learned something about triangular numbers and complete graphs. The complete graph on $n$ vertices is denoted by $K_n$. In our case $n=5$ so that the $K_5$ graph look like this:

The number of triads are equivalent to the number of connected edges, given by:

$r = {n \choose k} = {5 \choose 3} = 10$

Coincidentally, this is also equivalent to ${5 \choose 2}$, as was the case for the number of pairs. Also, notice how each vertex is connected to the other four.