What is the probability that a number chosen at random from the range to inclusive will contain ten different digits? Round your answer to the nearest millionth.

Source: NCTM Mathematics Teacher, January 2006

**Solution**

We divide the large range into nine smaller ranges:

to

to

to

to

to

We examine the first range in details.

The first digit after has nine choices

The second digit after has eight choices

The third digit after has seven choices, etc.

Number of desirable outcomes

Similarly, each of the other ranges has desirable outcomes.

Total number of desirable outcomes =

Total number of possible outcomes

Probability that a number chosen at random will contain ten different digits

**Answer**: