Editing Combinatorics (section)

== <span style="color: #FFFFFF;">Analyzing</span> ==
{| class="wikitable"
|+ Classical Combinatorial Results
! Result !! Statement !! Application
|-
| Pigeonhole Principle || n+1 objects in n boxes → repeat || Birthday paradox; hash collisions; proofs
|-
| Binomial Theorem || (a+b)^n = Σ C(n,k) a^k b^{n-k} || Probability; algebra; generating functions
|-
| Ramsey R(3,3)=6 || 6 people → 3 friends or 3 strangers || Guaranteed structure in large structures
|-
| Four Color Theorem || Every planar map 4-colorable || Map coloring; register allocation in compilers
|-
| Erdős-Rényi ER(n,p) || Phase transition at p=1/n || Random graph theory; network models
|}

'''NP-complete graph problems''': Graph 3-coloring, Hamiltonian cycle, Maximum clique, Vertex cover, Independent set, Subgraph isomorphism. These are all equivalent in hardness (polynomial-time reducible to one another) and are NP-complete — unlikely to have efficient algorithms.
</div>

<div style="background-color: #483D8B; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">