Problem (Combinatorics):
Three different numbers are chosen at random from the set $\{1,2,3,4,5\}$.

The numbers are arranged in increasing order.

What is the probability that the resulting sequence is an arithmetic sequence?

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3,5,7,9 is an arithmetic sequence with four terms.)

Solution:
We consider choosing the three numbers all at once.

We list the possible sets of three numbers that can be chosen:

$$
\{1,2,3\}\{1,2,4\}\{1,2,5\} \quad\{1,3,4\} \quad\{1,3,5\} \quad\{1,4,5\} \quad\{2,3,4\} \quad\{2,3,5\} \quad\{2,4,5\} \quad\{3,4,5\}
$$

We have listed each in increasing order because once the numbers are chosen, we arrange them in increasing order.

There are 10 sets of three numbers that can be chosen.

Of these 10, the 4 sequences 1,2,3 and 1,3,5 and 2,3,4 and 3,4,5 are arithmetic sequences. Therefore, the probability that the resulting sequence is an arithmetic sequence is $\frac{4}{10}$ or $\frac{2}{5}$.

Final answer: $\frac{2}{5}$