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Let $p_1,p_2,p_3\in\mathbb{P}^n$ be three general points, $X$ the blow-up of $\mathbb{P}^n$ at $p_1,p_2,p_3$, then along the lines $\left\langle p_i,p_j\right\rangle$, and finally along the plane $\left\langle p_1,p_2,p_3\right\rangle$, with exceptional divisors $E_i$, $E_{i,j}$ and $E_{1,2,3}$. Let $H$ be the pull-back of the hyperplane section.

How may we compute the intersection numbers $H^{k}\cdot E_1^{k_1}\cdot E_{2}^{k_2}\cdot E_{i,j}^{k_{i,j}}\cdot E_{1,2,3}^{h}$ with $k+k_1+k_2+k_{i,j}+h = n$?

Some of them are trivially zero. Using these and some other relations I got $E_i^n = (-1)^{n-1}$ and $E_{i,j}^n = 2\cdot (-1)^{n-1}$. Could we compute these numbers by looking at the normal bundles?

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Given a smooth projective variety $X$ and a smooth subvariety $Y$, there is a formula for the Chow Ring of the blow-up of $X$ along $Y$ in terms of the Chow Rings of $X$ and $Y$ and the Chern/Segre classes of the Normal bundle of $Y$ in $X$. (For example, have a look at Fulton's book on "Intersection Theory"). Applying this repeatedly should get you what you need.

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