What is the probability that a permutation of the ten digits

What is the probability that a permutation of the ten digits is in numeric order (09)(09)? The caveat is that at least one of the digits is in the proper place for the permutation, but we do not know which digit.

Solution

Probability that permutation of 10 digits (0-9) is in numerical order:

Number of ways in which (0-9) are in proper order / Total number of ways

Number of ways in which (0-9) are in proper order = 1

Total number of ways = 10P10 = 10! = 3628800

Probability = 1 / 10! = 1 / 3628800 = 2.7557 * 10^-7

Probability that permutation of 10 digits (0-9) is not in numerical order = 1 - (1/3628800)

= 0.9999997244