MathDB

Suppose that each of

n

people writes down the numbers

1, 2, 3

in random order in one column of a

3\times n

matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums

a, b, c

of the resulting matrix be rearranged (if necessary) so that

a \le b \le c

. Show that for some

n \ge 1995

,it is at least four times as likely that both

b = a+1

and

c = a+2

as that

a = b = c

Problem Statement