HMMT 二月 2009 · COMB 赛 · 第 7 题

HMMT February 2009 — COMB Round — Problem 7

[ 5 ] Paul fills in a 7 × 7 grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again, to obtain two different random arrangements of the numbers in the grid. What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?

解析

[ 7 ] Paul fills in a 7 × 7 grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again (to obtain two different random arrangements of the numbers in the grid). What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements? Answer: 147 / 2 7 ( ) 14 · ( ) 49 2 Solution: Each of the pairs of numbers has a probability of = 1 / 4 of being in the same 49 2 ( ) 2 row or column in one of the arrangements, so the expected number that are in the same row or column in both arrangements is ( ) 49 147 2 · (1 / 4) = . 2 2