Properties of Binomial Coefficients

Abstract

[AIME 1987] A given sequence r ₁, r ₂, ..., r _n of distinct real numbers can be put in ascending order by means of one or more “bubble passes.” A bubble pass through a given sequence consists in comparing the second term with the first term and exchanging them if and only if the second term is smaller, then comparing the third term with the current second term and exchanging them if and only if the third term is smaller, and so on, in order, through comparing the last term, r _n , with its current predecessor and exchanging them if and only if the last term is smaller. Figure 3.1 shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass.