HMMT 十一月 2013 · 团队赛 · 第 3 题

HMMT November 2013 — Team Round — Problem 3

[ 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 expected value of M · N . Power of a Point

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 expected value of M · N . Answer: 143745 By linearity of expectation and symmetry, 2 E [ M N ] = E [(100 A + 10 B + C )(100 D + 10 E + F )] = 111 · E [ AD ] . 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.