HMMT 二月 2000 · 几何 · 第 1 题

英文解析

1. Consider the b oard lab eled as b elo w, with lab els for olumns and ro ws. T o ho ose an yre tangle on the b oard, it is suÆ ien t to ho ose some number (1-8) of adja en t olumns,
   and some number (1-8) of adja en t ro ws, sin e the re tangle an be reated b y formingthe in terse tion of the olumns and ro ws. F or instan e, the in terse tion of olumns 2,3
   and ro ws 3,4,5 is the re tangle shaded b elo w. So, there are 8 w a ys to ho ose 1 adja en tolumn, 7 w a ys to ho ose 2 adja en t olumns, : : : , 1 w a y to ho ose 8 adja en t olumns,
   so there are 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36 total w a ys to ho ose the olumns, and
   36 w a ys to ho ose the ro ws. Th us, the total number of w a ys to ho ose a re tangle (i.e.
   the total number of re tangles) is 36 = 1296 .2
   5678
   1234
   1 2 3 4 5 6 7 8