HMMT 二月 2009 · COMB 赛 · 第 7 题
HMMT February 2009 — COMB Round — Problem 7
题目详情
英文原题
- [ 5 ] Paul fills in a 7 × 7 grid with the numbers 1 through 49 in a random arrangement. He then eraseshis work and does the same thing again, to obtain two different random arrangements of the numbersin the grid. What is the expected number of pairs of numbers that occur in either the same row aseach other or the same column as each other in both of the two arrangements?
解析
英文解析
- [ 7 ] Paul fills in a 7 × 7 grid with the numbers 1 through 49 in a random arrangement. He then eraseshis work and does the same thing again (to obtain two different random arrangements of the numbersin the grid). What is the expected number of pairs of numbers that occur in either the same row aseach other or the same column as each other in both of the two arrangements?
Answer: 147 / 2
( )7
14 ·
( )
Solution: Each of the pairs of numbers has a probability of = 1 / 4 of being in the same 249
( )249
row or column in one of the arrangements, so the expected number that are in the same row or column 2
in both arrangements is
( )
49 147
· (1 / 4) = .2
2 2