An extension of the so-called committee problem is discussed in terms of the occupancy of a certain matrix whose row totals are taken as fixed, and whose cell entries are constrained to lie between zero and defined maxima. A statistic is constructed as a function of the column totals and used to test the significance of a column effect, irrespective of row differences. Two special cases when the cell maxima are constant, first within rows and second within columns, yield a covariance matrix for the column totals whose inverse may be found explicitly, and this in turn leads to a relatively simple expression for the test statistic. Applications of this generalized problem to certain health research situations are discussed, and the adequacy of a X2 approximation to the statistic's null distribution is mentioned.