(Betti numbers)
Let \(X_\Sigma\) be smooth and complete of dimension \(n\), and write \(\Sigma(k)\) for the set of \(k\)-dimensional cones. Then the odd Betti numbers vanish and \[ \beta_{2k} = \sum_{i=k}^{n} (-1)^{i-k} \binom{i}{k} \size \Sigma(n-i) . \]
(Two special cases)
\[ \beta_0 = \beta_{2n} = 1, \qquad \chi(X_\Sigma) = \size \Sigma(n) , \] and when every maximal cone has dimension \(n\), \[ H^2(X_\Sigma; \ZZ) \cong \Pic(X_\Sigma), \qquad \beta_2 = \size \Sigma(1) - n . \]
(Where the formula comes from)
Cohomology of an affine piece is exterior algebra on the lattice orthogonal to the cone, \[ H^i(U_\sigma; \ZZ) \cong \Extpower^i M(\sigma), \qquad M(\sigma) \da \sigma^\perp \intersect M , \] so the Mayer–Vietoris spectral sequence for the cover by the \(U_\sigma\) over maximal cones has \[ E_1^{p,q} = \bigoplus_{i_0 < \cdots < i_p} \Extpower^q M(\sigma_{i_0} \intersect \cdots \intersect \sigma_{i_p}) \abuts H^{p+q}(X_\Sigma; \ZZ) . \] The alternating sum over \(q\) of \(\rank \Extpower^q M(\tau)\) is \(0\) unless \(\dim \tau = n\), where it is \(1\), and only the maximal cones survive. That is the Euler characteristic statement, and tracking the individual ranks gives the binomial formula.
(Check it on the two standard surfaces)
For \(\PP^2\): \(\size \Sigma(2) = 3\), \(\size \Sigma(1) = 3\), \(\size \Sigma(0) = 1\), giving \(\beta_0 = 1\), \(\beta_2 = 1\), \(\beta_4 = 1\) and \(\chi = 3\). For \(\FF_a\): \(\size \Sigma(2) = 4\), \(\size \Sigma(1) = 4\), giving \(\beta_2 = 2 = \rank \Pic(\FF_a)\) and \(\chi = 4\).
The formula turns cohomology into cone counts, while the torus-orbit cell decomposition explains structurally why the odd cohomology vanishes.