The number of m-faces of an n-cube

n-cube is a generalization of a cube where 0-cube is a vertex, 1-cube is a line segment, 2-cube is a square, 3-cube is a cube and 4-cube is a hypercube. n-cube has its m-faces where 0-faces are vertexes, 1-faces edges and 2-faces are faces.

Ever wondered how many m-faces does an n-cube have? It is easy to calculate with the following hint as a starting point. Consider a cube with points \{(0,0,0),(0,0,1),(0,1,0),(1,0,0),(0,1,1),(1,0,1),(1,1,0),(1,1,1)\}. What is the necessary and sufficient condition on the coordinate values of the four points of a cube to form a face?

Afterwards, you should obtain a formula:

The number of m-faces of n-cube is {n \choose m} 2^{n-m}.

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