# Twin Simplex

Geometry Level 5

Let $$e_1, \ldots , e_{n+1}$$ be the standard basis in $$\mathbb R^{n+1}$$. We define an $$n$$-dimensional simplex as the convex hull of the endpoints of the vectors $$e_1, \ldots , e_{n+1}$$. That is, the $$n$$-dimensional simplex can be defined as the convex hull of $$n + 1$$ affinely independent points.

We can thus clearly see that the standard 1-simplex is a line segment, the standard 2-simplex is an equilateral triangle, the standard 3-simplex is a regular tetrahedron, and so on.

How many 4-dimensional faces does the 197-dimensional simplex have?

Bonus: Generalize the problem. How many $$k$$-dimensional faces does an $$n$$-dimensional simplex have?

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