The entries of (*)
Erdos' squarefree conjecture asks whether the central one of these numbers is always squarefree,and they are the subject of Vandermonde's convolution formula. The Catalan numbers can bedefined using these numbers of indices 2n and n times one over n plus one. The hockey stick identitycan be used to add them, and summing over their lower index from 0 to n gives two to the nth power.The entries of (*) Pascal's triangle are these, which arise when multiplying a two-term polynomial withitself. These numbers are symbolized by a column matrix with two entries, and they are equal to n factorialdivided by r factorial times n minus r factorial. For 10 points, name these objects, which give the number ofways to select r objects from a group of n.
Erdos' squarefree conjecture asks whether the central one of these numbers is always squarefree,and they are the subject of Vandermonde's convolution formula.
 binomial coefficients [or n choose r or combinations or entries in Pascal's Triangle or equivalents before mention; do NOT accept "permutations"]