Number of faces of dimension p of simplex

Question

How can I prove that the number of faces of dimension p of an an n-dimensions simplex is represented by the binomial coefficient below?

${n+1}\choose{p+1}$

Answer 1

an $n$-dimensional simplex is made up by $(n+1)$ vertices. any $(p+1)$ vertices will make up one $p$-dimensional face. (it takes two vertices to create an edge, three to make a triangle, 4 for a tetrahedron, etc...) in fact, there is precisely a $p$-simplex between any $(p+1)$ vertices, because simplexes are complete graphs.