Information is available from a diallel cross consisting of all possible crosses among n parental lines, excluding the parental lines themselves and assuming that reciprocal crosses are identical. The lines are assumed to be a random sample from an infinite population of such lines. The general combining abilities of the lines may be more important than the specific combining abilities of the individual crosses. If so, we may decide to rank the lines on the basis of their estimated general combining abilities and make crosses for further testing, or for commercial production, using only lines with high estimated general combining abilities. Existing statistical selection theory can be used to predict the expected genetic superiority of these crosses. For n = 6, 10, and 20, the best five crosses would be expected to result from crossing the lines as follows, where the line number is its rank order on estimated general combining ability: 1 × 2, 1 × 3, 1 × 4, 2 × 3, 1 × 5. Crosses 2×3 and 1 × 5 may not necessarily be expected to be in the stated order, depending on the value of n. In a particular trial, the estimated general combining abilities may suggest that other crosses should be made, but the practical importance of this is doubtful. Note that if only three crosses are to be selected, 1 × 4 is expected to be better than 2 × 3, i.e., the best three crosses are not obtained by crossing the best three lines. This method of selection based on general combining abilities will be referred to as line selection.