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?

Answer 1

The set is there to indicate that the three numbers chosen are distinct.
Altogether there are 6 ways to arrange 3 distinct numbers (=3!), out of which an arithmetic sequence is either
a,b,c
or
c,b,a
assuming a<b<c.
Can you calculate the probability?

Answer 2

An arithmetic sequence is one in which the differences between neighboring numbers is the same. The only arithmetic sequences you can get from that set of 5 numbers are: (1, 2, 3), (2, 3, 4), (3, 4, 5), or (1, 3, 5). So only 4 choices.

How many ways can you arrange 3 numbers from the set so that they are in increasing order?

123, 124, 125,
134, 135,
145,
234, 235,
245,
345.

You should be able to find the probability. Can you argue why you don't have to worry about choices like 524 and so forth?