It is more than 91 years since, on January 17th 1871, a group of twenty-six men met in another place and formed the Association for the Improvement of Geometrical Teaching, which twenty-five years later became the Mathematical Association. Most were schoolmasters; some were clergymen; the account of the proceedings in the first annual report adds: “There were also a few gentlemen present who were not members of the Association.” The ladies were allowed to join later.

In a great many problems in probability, one is given one set of quantities Xj expressed in terms of a second set of quantities Yk, and one needs an expression for the Yk in terms of the Xj. Basically, of course, this is just a problem in solving equations, but the results can be expressed in a particularly elegant form when the coefficients involve the binomial quantities . This situation is one that occurs very frequently, and the resulting formulae simplify the derivations of many identities. The present expository account is intended to collect together various results in a convenient form. We shall need to recall briefly a few of the main properties of the binomial coefficients.

The rank of the coefficient matrix plays a dominant role in the theory of linear algebraic equations. It is not surprising, therefore, that a test for the rank of a matrix, that was a by-product of some work in dimensional analysis, proves to be an an admirable tool in this theory With its aid the consistency requirement assumes a simple and effective form, and the solution of both homogeneous and non-homogeneous systems is given explicitly in terms of submatrices.

The following is offered as an example of a proof in which the important part is most easily written out in the form of a block flow diagram familiar to programmers of universal computers.

We shall be concerned with matrices of only a particular kind, namely antisymmetric matrices whose non-diagonal elements are all plus or minus unity. Call such matrices, p-matrices.

The American Mathematical Monthly in Vol. 68 (1961) page 380 gave a solution by Leon Bankoff of the following problem proposed by A. Oppenheim of the University of Malaya in Vol. 67 (1960) page 802.