Tina randomly selects two distinct numbers from the set $$\{1,2,3,4,5\},$$and Sergio randomly selects a number from the set $$\{1,2,\ldots,10\}.$$What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
There are ten ways for Tina to select a pair of numbers. The sums 9, 8, 4, and 3 can be obtained in just one way, and the sums 7, 6, and 5 can each be obtained in two ways. The probability for each of Sergio's choices is $1/10$. Considering his selections in decreasing order, the total probability of Sergio's choice being greater is \begin{align*}
&\left(\frac{1}{10}\right)\left(1 + \frac{9}{10} + \frac{8}{10} +
\frac{6}{10} + \frac{4}{10} + \frac{2}{10} + \frac{1}{10} + 0 + 0 + 0
\right) \\
& = \boxed{\frac{2}{5}}.
\end{align*}