HMMT 十一月 2013 · 团队赛 · 第 3 题
HMMT November 2013 — Team Round — Problem 3
题目详情
英文原题
- [ 6 ] The digits 1 , 2 , 3 , 4 , 5 , 6 are randomly chosen (without replacement) to form the three-digit numbers
M = ABC and N = DEF . For example, we could have M = 413 and N = 256. Find the expectedvalue of M · N .
Power of a Point
解析
英文解析
- [ 6 ] The digits 1 , 2 , 3 , 4 , 5 , 6 are randomly chosen (without replacement) to form the three-digit numbers
M = ABC and N = DEF . For example, we could have M = 413 and N = 256. Find the expectedvalue of M · N .
Answer: 143745 By linearity of expectation and symmetry,
E [ M N ] = E [(100 A + 10 B + C )(100 D + 10 E + F )] = 111 · E [ AD ] .2
Since
2 2 2 2 2 2 2
(1 + 2 + 3 + 4 + 5 + 6) − (1 + 2 + 3 + 4 + 5 + 6 ) 350
E [ AD ] = = ,
6 · 5 30
our answer is 111 · 35 · 37 = 111 · 1295 = 143745.