The largest possible number in the collection is 18.
In order to have a mean of 10, the sum of all ten numbers must be 100. Additionally, with the range equal to 10, the maximum number in the set is 10 more than the minimum.
The minimum number in the collection must be less than 10. Otherwise, to have a range of 20, the sum of the ten numbers would have to exceed 100.
Suppose that 9 was the minimum and 19 the maximum. The middle eight numbers would then have to have a sum of 72. But the middle eight numbers are all at least 9 and could only have a sum of 72 if they were all equal to 9. If that were the case, then the set would have a median of 9. Thus the minimum number in the collection must be less than 9 (and therefore, the maximum must be less than 19).
The collection below shows that it is possible to have 8 as the minimum and 18 as the maximum as this collection satisfies all four conditions. We therefore, conclude that 18 is the largest possible number in such a collection.