How the number of binary collisions increases with centrality faster than the number of participant in heavy ion collisions at different particle colliders?
It's combinatorics, a variant of the "handshake problem." Suppose you are at a party with $N$ other people and you want to shake everyone's hand: you're going to give $N$ handshakes. But so is everyone else, so the total number of
binary interactions handshakes will be $\frac12N(N+1)\approx N^2/2$. You can do this by hand for small parties:
- 1 person: no handshakes
- 2 people: 1 handshake
- 3 people: 3 handshakes (AB, AC, BC)
- 4 people: 6 handshakes (AB, AC, AD, BC, BD, CD)
- 5 people: 10 handshakes
In collisions you have the additional complication of pair creation in some binary collisions increasing the number of participants as well.