Question

In a lottery 5 numbers and chosen from the set {1,...,90} without replacement. We order the five numbers in increasing order and we denote by X the number of times the difference between two neighboring numbers is 1. (E.g. for {1,2,3,4,5} we have X=4, for {3,6,8,9,24} we have X=1, and for {12,14,43,75,88} we have X=1. ) Find EX.

Answer 1

Let the be the index number random variables representing difference of digits of sorted numbers at positions . If the digits differ by 1, then , other wise 0.

Let the digits at 1,2 be . Then in the sorted numbers the remaining 3 digits can be filled in ways.

So,

Let the digits at 2,3 be . then

Let the digits at 3,4 be . then

Let the digits at 4,5 be . then

Now   the number of times the difference between two neighboring numbers is 1 is the random variable,

By the property of linearity of expectation, (note is either 1 or 0)